Course webpage for MA 541, Modern Algebra I, Fall 2021

Archived version (all solution sets and quiz/exam links are dead)

Instructor: Anna Medvedovsky

Textbook: Tom Judson, Abstract Algebra: Theory and Applications, annual edition 2021, available free online or as a paper copy for purchase.



Class meetings and homework assignments

References are to Judson unless otherwise specified.

Date Lecture Topics Reading Homework

Statement of the structure theorem for finite abelian groups (see Section 13.1): each finite abelian group is a product of finite abelian p-groups, and each finite abelian p-group is in turn a product of cyclic abelian p-groups.

Platonic solids and their rotational symmetries. There are five platonic solids: the tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron. Duality between the cube and the octahedron and between the dodecahedron and the icosahedron. The group of rotational symmetries of the cube or octahedron is S4; the isomorphism can be realized through the action on the four vertex-to-vertex diagonals of the cube (see Theorem 5.27). The group of rotational symmetries of the icosahedron or dodecahedron is A5. This may be realized as the faithful action on five cubes inscribed in the dodecahedron whose edges are diagonals of face pentagons (here’s one such cube); alternatively, each cube can be seen in the icosahedron, with the six faces parallel to six well-choisen edges. Or this may be realized as the faithful action on five tetrahedra inscribed in the dodecahedron, which can also be seen in groups of four well-chosen faces of the icosahedron.

In fact, one can show that the only finite rotation groups in 3 dimensions are cyclic, dihedral, or correspond to rotational symmetries of one of the platonic solids: A4, S4, or A5.


p-groups: A nontrivial group is a p-group for some prime if its every element has p-power order. A finite group is a p-group if and only if it has pn elements for some n > 0. A p-Sylow subgroup of a finite group G of order pkm (here m is prime to p and k ≥ 1) is a subgroup of G of order pk.

The Sylow theorems (see section 15.1 describe the p-Sylow subgroups of a group G if p divdes |G|: p-Sylow subgroups always exist, they are all conjugate, and their number is both congruent to 1 modulo p and divides the size of the group. See this set for the formulation of the Sylow theorems given in class. See also Prof. Conrad’s notes on the Sylow theorems.

Brief interlude on conjugate subgroups: if H is a subgroup of a group G containing g, then gHg–1 is also a subgroup of G, and the conjugation-by-g map H → gHg–1 is an isomorphism of groups.

Examples of p-Sylow subgroups for S3, Q8, D6, and S4.

We can use the Sylow theorems both to classify all groups of certain orders n up to isomorphism (see, for example, (2) and (3) here) and to prove that no group of order n is simple (which we did in class for 17 ≤ n ≤ 30). How do these arguments fail for n = 60?

More Sylow theorem problems for your after-the-semester pleasure!


Consequences of group actions for the structure of groups.
  • Class equation (Section 14.2): If G is a finite group, then
    |G| = Z(G) + [G : Z(x1)] + ... + [G : Z(xk)],
    where x1, ..., xk in G are representatives of the non-singleton (or non-central) conjugacy classes of G and Z(xi) is the centralizer of xi, the subgroup of elements of G that commute with xi.
  • Lemma: If a group of order pn (p prime, n ≥ 1) acts on set X, then
    |X| ≡ |Fixed points| modulo p.
    See Theorem 4.1 of Conrad’s excellent notes on group actions.
  • Cauchy’s theorem (Theorem 15.1): any finite group whose order is divisible by p must have an element of order p. (In fact, there are at least p – 1 of them — why?)
  • A group of order pn for n ≥ 1 always has a nontrivial center (Theorem 14.15).

The search for simple groups: We use the following series of lemmas to show that the only simple groups of order ≤ 16 are cyclic groups of order p. Here p is always a prime.
  • Lemma P: Zp is a simple group.
  • Lemma Ab: The only abelian simple groups are the cyclic groups of order p.
  • Lemma 2P: No group of order 2p is simple. Proof: Any such group is either cyclic or dihedral (Lecture 20).
  • Lemma P2: No group of order p2 is simple. Proof: HW #8 (9c).
  • Lemma Z: The only simple groups with nontrivial center are cyclic groups of order p. Proof: The center Z(G) is always normal in G. If Z(G) = G, then G is abelian; use Lemma Ab.
  • Lemma PQ: No group of order pq, where p and q are both prime, is simple. Proof: Suppose q ≥ q. Cauchy’s theorem guarantees a subgroup of order q, which then has index p. Since p is the smallest prime dividing |G|, this subgroup is normal (HW #8 (8)).
  • Lemma Pn: The only simple groups of order pn are the cyclic groups of order p. Proof: a group of order pn always has a nontrivial center (Theorem 14.15). Now use Lemma Z.
  • Lemma R!: If G has a conjugacy class of size r > 1 and |G| > r!, then G is not simple. Proof: The conjugation action of G on this conjugacy class gives a homomorphism of G to Sr. This homomorphism is not injective because |G| > |Sr|. This homomorphism has nontrivial image because the action is transitive and r > 1. Therefore the kernel of this homomorphism is a nontrivial proper subgroup of G.

More on quotient groups: Fix a normal subgroup N of a group G throughout. We proved that the binary operation G/N × G/N → G/N given by (aNbN) mapsto abN is well defined and gives a group structure on G/N with which the natural projection map πN : G → G/N sending a to aN is a surjective group homomorphism with kernel N (see also section 11.2).

First isomorphism theorem (Theorem 11.10):
  • Statement: If f : G → H is a homomorphism of groups, then N = ker f is a normal subgroup of G, and K = im f is a subgroup of H, and the map f˜ : G/N → K induced by f given by f˜(aN) := f(a) is an isomorphism of groups.
  • Example: f : R → C× mapping x to ex. We have ker f = Z and im f = T, so f induces an isomorphism R/Z ≅ T.
  • Example: f : R2 → R mapping (ab) to 2a – b. We have ker f = (1, 2)R and im f = R, so f induces an isomorphism R2/(1, 2)R ≅ R.
  • Corollary: If G is a finite group and f : G → H is a group homomorphims, then |G| = |ker f| |im f|.
  • This corollary is completely analogous to the rank-nullity theorem from linear algebra: if V and V are two R-vector spaces and f : V → W is an R-linear map between them, then dim V = dim ker f + dim im f.
Universal property of quotient groups: If f : G → H is a group homomorphism whose kernel contains N, then f “factors through G/N”, in the sense that there is a unique group homomorphism α : G/N → H satisfying f = α ⚬ πN.

More properties:
  • Correspondence theorem (Theorem 11.13): There is a inclusion-preserving bijection between
    {subgroups of G/N} ↔ {subgroups of G containing N}
  • sending a subgroup K of G/N to πN–1(K) and a subgroup H of G containing N to πN(H) = H/N.
  • This correspondence sends normal subgroups to normal subgroups. If H is a normal subgroup of G containing N, then there is a natural isomoprhism (G/N)/(H/N) ≅ G/N sending aN(H/N) to aH. This is the third isomorphism theorem (Theorem 11.14).
  • Second isomorphim theorem (Theorem 11.12): If H is any subgroup of G, then HN is a subgroup of G containing N as a normal subgroup, and HN is normal in N, and there is a natural isomorphism HN/N ≅ H/(HN) sending hnN to h(HN).
The Hölder program and the search for finite simple groups:
  • If a finite group G has a normal subgroup N, then N and G/N are two smaller groups that fit together to make up G. We can continue this process and look for normal subgroups of N and G/N until we reach simple groups (nontrivial groups with no nontrival proper normal subgroups).
  • The Jordan-Hölder theorem (Theorem 13.18) makes this precise: every finite group G has a composition series (a descreasing sequence of nested subgroups, each normal in the previous one, with simple quotients). This composition series need not be unique but the and that the multiset of simple quotients is uniquely determined by G.
  • One approach to understanding all finite groups is summarized by the Hölder program: to understand all finite groups we must (1) understand all finite simple groups and (2) understand how finite simple groups can fit together.
  • Part (1), the search for all finite simple groups is a gargantuan task that took up the entire 20th century of group theory efforts. The final classification theorem was completed in 2004. In our last few classes we will try to capture some of the early excitment of the hunt.
No more!

Normal subgroups and quotient groups, also known as factor groups: see Section 10.1.

A subgroup H of a group G is normal if any of the following equivalent conditions are satisfied (see also Theorem 10.3):
  • For all g in G, h in H we have ghg‑1 in H.
  • For all g in G we have gHg‑1 ⊆ H.
  • For all g in G we have gHg‑1 = H.
  • For all g in G we have gH = Hg.
  • The partition of G into the left cosets of H is the same as the partition of G into the right cosets of H.
  • H is a union of conjugacy classes of elements of G
  • H is the kernel of a group homomorphism from G to some group.
Examples of normal subgroups: any subgroup is normal in an abelian group; the trivial subgroup is normal in any group; any group is normal as a subgroup of itself; An is normal in Sn; more generally, any subgroup of index 2 is normal; the center of a group is a normal subgroup of that group; more generally, the kernel of any group action is a normal subgroup of the group.

Digression: Consider the action of a group G on itself by conjugation (see also Section 14.2).
  • The orbit of an element x of G is called its conjugacy class. In an abelian group, every conjugacy class is a singleton.
  • This action is essentially never transitive (it’s only transitive when G is the trivial group).
  • The stabilizer of any x in G is the set of elements that commutes with x: this subgroup is called the centralizer of x, denoted Z(x) or ZG(x).
  • The kernel of the action (this is also the intersection of all the stabilizers) is the set of elements that commute with every element of G: this is precisely the center of G, denoted Z(G). (This formulation shows that the center is a normal subgroup of G.)
  • The orbit-stabilizer formula in this case implies that if G is finite, then the size of each conjugacy class is a divisor of G. Since the conjugacy classes also partition G, and since the identity is always in its own conjugacy class, this is a very limiting condition! For example, if G has order 6, then there are only two possibilities: 6 = 1 + 1 + 1 + 1 + 1 + 1 and 6 = 1 + 2 + 3. The first is realized by Z6 and the second by S3.
Theorem (see Theorem 10.4): If N is a normal subgroup of a group G, then the coset space G/N inherits a natural group structure from G under the operation aN ⚬ bN := abN. With this group structure the map G → G/N sending a in G to aN is a surjective group homomorphism with kernel N.
  • Alternatively, if N is a normal subgroup, one can show that the set of pairwise products (aN)(bN) = {xy : x in aNy in bN} forms a coset of N in its own right, namely the coset abN. This pairwise product multiplication structure is an equivalent way of defining the group operation on G/N.
Exampes: (1) Z/3Z = Z3. (2) The quotient Sn/An has size 2, so is isomorphic to {1, ‑1}. This isomorphism realized by the sign map. (3) The center of D4 is Z(D4) = {1, r2}, where r is the rotation by 90 degrees as usual. Therefore D4/Z(D4) is a group of order 4. The only elements of D4 of order 4 are r and r3; since their square is in Z(D4), the images of both of these have order 2 in D4/Z(D4). Therefore the quotient group D4/Z(D4) is isomorphic to the Klein-4 group.

HW #8

(due 11/30/21,
12/2/21 in class ok)

LaTeX file

Fundamental theorem of group actions again. Proof of the fact that elements in the same orbit have conjugate stabilizers. Proof of the third part: natural bijection between G/Stab(x) and Orb(x) that is compatible with the G-action on both (in other words, these are isomorphic as G-sets).

Action of D3 on vertices of triangle, edges of triangle (same as vertices as the equilateral triangle is self-dual), points of the triangle. Action on vertices is faithful and transitive, and gives an isomorphism D3 ≅ Perm(vertices) = S3. Action on the points of the triangle has an orbit of size 1 (the center) and stabilizer D3, points with orbits of order 3 (any points on the flip axes except for the center) and stabilizers of order 2, and points with orbits of order 6 (all the other points) and trivial stabilizer.

Group R of rotations of a regular tetrahedron, isomorphic to A4, acts on vertices (transitive and faithful action, stabilizers have order 3, yields the homomorphism R ↪ Perm(vertices) = S4), acts on faces (same story as the action on vertices as the regular tetrahedron is self-dual), acts on edges (transitive and faithful action, stabilizers have order 2, yields a homomorphism R ↪ Perm(edges) = S6). [Can you show that the last map actually lands in A6? Do the same analysis of orbits and stabilizers we did for D3 for the action of R on the points of the tetrahedron.]

Group T of all symmetries of a regular tetrahedron contains R as an index-2 subgroup; action on vertices leads to the isomorphism T ≅ Perm(vertices) = S4.
problems for HW #8
(typo fixed 11/23)

HW #8 in full

Group actions (section 14.1 or Prof. Conrad’s notes). Definitions. First examples: trivial action of any group on any set; Sn acts on {1, 2, ..., n} by permutations; more generally subgroups of Perm(X) act on a set X by permutations; GL2(R) acts on R2 by linear transformations; Dn acts on the vertices/edges/set of points of a regular n-gon or even on the entire plane; any subgroup H of a group G acts on G by left translation; a group G acts on the coset space G/H for any subgroup H by left translation, a group G acts on itself by conjugation (here g · x = gxg‑1).

Lemma: if X is a G-set, then for any g in G, the map X to X sending x to g · x is a permutation of X. Theorem: The action of G on a set X induces a homomorphism G → Perm(X), and vice versa. See Theorem 1.6 of Conrad notes.

More definitions: for a G-set X and x in G, the orbit Orb(x) is the subset {g · x : g ∈ G} of X and the stabilizer Stab(x) is the subset {g ∈ G : g · x = x} of G. The point x is a fixed point of the action if Stab(x) = G. The action is transitive if there is exactly one orbit (in class, I forgot to stress that by definition we do not consider the action on the empty set transitive!). The action is faithful if every nonidentity element of G moves something: that is, the corresponding homomophism G → Perm(X) is injective.

Fundamental theorem of group actions (see Theorem 3.16 of Conrad notes): Let X be a G-set.
  • The orbits of the action partition X.
  • The stabilizers of the action are subgroups of G and
    Stab(gx) = g Stab(xg‑1.
  • For any x ∈ X, there is a natural bijection between the coset space G/Stab(x) and the orbit space Orb(x) sending g Stab(x) to g · x. Moreover, this bijection is compatible with their respective G-actions.
Corollary (orbit-stabilizer formula): For any any x ∈ X, we have |Orb(x)| = [G : Stab(x)]. In particular, if G is finite, then the length of any orbit always divides the size of the group.

If X is a G-set...
  • Show that the relation
    x ∼ y if there exists g in G so that g · x = y
    is an equivalence relation on X.
  • Conclude that the orbits of the action partition X.
  • Show that stabilizers of elements of X are subgroups of G.
  • Show that elements in the same orbit have conjugate stabilizer subgroups.

  • Think about the orbits of various points in the plane under the action of Dn.

    More on the dihedral group Dn of order 2n for n ≥ 3 (section 5.2): the n rotations form a cyclic subgroup Rot(n) of order n; let r be a generator. This subgroup has index 2, so that the n reflections form the non-identity coset of Rot(n) in Dn. If f ∈ Dn – Rot(n) is any reflection, then Dn is generated by r and f, with the fact that (fr)2 = 1 (because fr is also a reflection) equivalent to their commutation relation fr = r–1f, so that a presentation for Dn is ⟨RF | Rn = F2 = 1, FR = R–1F⟩. We can then also extend the definition of Dn to n = 1 and n = 2: D1 (symmetries of a fat point that can be flipped) is the group of order 2 and D2 (symmetries of a fat segment) is the Klein-4 group.

    Theorem: If G is a group of order 2p for p a prime, then G is either cyclic or dihedral.

    Proof: If p = 2, we already know this result about groups of order 4, so assume p is odd. Suppose G is not cyclic, so there’s no element of order 2p. Then by a corollary to Lagrange’s theorem, the order of each nonidentity element is 2 or p. If every nonidentity element has order 2, then, taking two distinct such elements x and y, we can show that the set {1, x, y, xy} forms a subgroup of G of order 4 — contradicting Lagrange’s theorem, as 4 doesn’t divide 2p. Thus G has at least one element a of order p. We claim that every element of G – ⟨a⟩ has order 2: indeed, if G has an element z not in a of order p, then all the aizj for i, j in {0,... p – 1} are all distinct (see also (2) on
    HW #7) — but there’s no room for p2 distinct elements in G! Therefore every element of G – ⟨a⟩ has order 2, and the same argument as above tells us that G = ⟨ab⟩ with an = b2 = 1 and ba = a–1b: in other words, that G is dihedral.

    Midterm return. Solutions.

    HW #7
    (due 11/18/21)

    LaTeX file

    Corollaries of Lagrange’s theorem — Theorem 6.10: if G is a finite group and H is a subgroup, then |G| = |H| [G:H].
    1. If G is a finite group and H is a subgroup, then |H| | |G| (Corollary 6.11).
    2. If G is a finite group and aG, then |ord(a)| | |G|.
    3. If G is a finite group and aG, then a|G| = 1. Compare to problem (7d) on HW #6, which asks for a different proof of this fact, without using Lagrange’s theorem, for abelian G.
    4. If G is a group of prime order p, then G ≅ Zp (Corollary 6.12).
    5. Fermat’s Little Theorem: If p is prime and a is an integer not divisible by p, then ap ‑ 1 ≡ 1 modulo p. Equivalently, if p is prime and a is any integer, then ap ≡ a modulo p (Corollary 6.19).
    6. Euler’s theorem: If a is relatively prime to a positive integer n, then aφ(n) ≡ 1 modulo n (Theorem 6.18).
    Euler’s theorem undergirds the RSA cryptosystem: see section 7.2.

    Dihedral groups (section 5.2): Fix n ≥ 3. The dihedral group of order 2n, denoted in this class by Dn, is the group of plane symmetries of a regular n-gon. (Beware! Sometimes the notation D2n is used instead. Both notations are in active use.) By our convention, D3 = Symm(▵) and D4 = Symm(□). The group Dn consists of n rotations, which form an index-2 subgroup of rotational symmetries, and n flips, either about n axes connecting vertices to midpoints of opposite sides (if n is odd) or about n/2 axes connecting pairs of opposite vertices and n/2 axes connecting midpoints of opposite sides (if n is even). To be continued...

    Two major topics coming up after we finish milking Langrange’s theorem: group actions (section 14) and quotient groups (also called factor groups, section 10.1).


    Cosets: Right and left cosets of a subgroup H of a group G. Examples: cosets of R(1, 2) in R2, cosets of 3Z in Z, cosets of Z in R, right and left cosets of ⟨(1 2)⟩ in S3, right and left cosets of A3 in S3.

    Cosets partition: The (left) cosets of H partition G (Theorem 6.4). To see this, either prove directly that two cosets are either disjoint or coincide, or interpret the equivalence relation from problem (6a) on HW #3 as the relation of being in the same left coset for H: for a, b in G we have a‑1b ∈ H iff a‑1bH = H iff aH = bH iff (WLOG) a ∈ bH (see also Lemma 6.3). An analogous argument works for right cosets; see problem (6d) on HW #3 for the equivalence relation.

    Lagrange’s theorem (Theorem 6.10): In a finite group, the order of any subgroup divides the order of the group. For the proof, you need a bijection between the elements of H and the elements of any other coset of H in G: see Proposition 6.9 or problem (7a) on HW #6. Corollary (Corollary 6.11): In a finite group, the order of any element divides the order of a group.

    Application: Fermat’s Little Theorem (Corollary 6.19): if p is a prime, then for any integer a not divisible by p, we have ap ‑ 1 ≡ 1 modulo p.


    Well-definition of the sign of a permutation redux: The sign of a permutation also appears in an expression for the determinant of an n × n matrix: if A = (aij)i,j=1...n, then
    det A = Σσ∈Sn sgn(σ)Πi=1...n ai, σ(i).

    The group of rotations of a regular tetrahedron has 12 elements, and by considering the action of the group on the 4 vertices, we can show that it is isomorphic to A4.

    Subgroup diagram of A4: four subgroups of order 3 generated by 3‑cycles, one Klein‑4 subgroup consisting of the pairs of 2‑cycles plus the identity, and three subgroups of order 2 generated by these pairs of 2‑cycles. (Exerices: why are these the only subgroups?)


    Cayley’s theorem: Every group is a subgroup of a permutation group (Theorem 9.12). Proof: the map G → Perm(G) sending g in G to the left-translation-by-g permutation tg satisfying tg(x) = gx is an injective group homomorphism. The map g ↦ [right-translation-by-g] doesn’t work. (Why not?)

    Transpositions Alternating groups
    • The set of even permutations form a subgroup of Sn denoted An and called the alternating group on n letters.
    • If n ≥ 2, then set of odd permutations is in bijection with An via right-translation-by-(1 2) map. The left-translation-by-(1 2) map also works. Therefore there are exactly as many odd permutations as even ones, and |An| = |Sn|/2 = n!/2.
    • More generally, if H is a subgroup of a group G, subsets of G of the form gH or Hg for various g in G are called (left or right) cosets of H in G. (Show that any coset of H is in bijection with H itself. What else do you need to conclude that if G is finite then |H| must divide |G|? Can you find a natural correspondence between the set of left cosets of H in G and the set of right cosets of H in G?)
    HW #6
    (due 11/9/21)

    LaTeX file

    Permutation groups (chapter 5)
    • For a set X, the set of bijections X → X forms a group, denoted Perm(X) or SX. If X and Y are sets of the same cardinality (that is, there is a bijection f : X → Y), then Perm(X) and Perm(Y) are isomorphic. (Why? Give an explicit isomorphism in terms of f.) So to understand Perm(X) for finite sets X we just consider Sn := Perm({1, 2, ..., n}) for n ≥ 1.
    • Sn has order n!. S3 is isomorphic to Symm(▵) by labeling the locations of the vertices of the triangle. By doing the same thing with the square, can view Symm(□) as a proper subgroup of S4. (In how many different ways can we do this?)
    • Cycle notation: definition of cycle, k-cycle (Judson calls these “cycles of length k”), disjoint cycles. Disjoint cycles commute (Proposition 5.8). A k-cycle has order k. The group Sn has “n choose k” times (k – 1)! k-cycles.
    • Theorem: every permutation in Sn is a product of disjoint cycles. (Why is σi as defined in class a cycle? Why is the decomposition of σ into a product of disjoint cycles unique up to reordering?) This is Theorem 5.9, but Judson doesn’t use the terminology of orbits or the equivalence relation imposed by an element σ of Sn (or, more precisely, the subgroup ⟨σ⟩) on {1, 2, ..., n}. This terminology instead first appears in section 14.1 on group actions.
    Coming up: expressing permutations as products of transpositions and the alternating groups, subgroups of A4, symmetries of a tetrahedron, dihedral groups, and the terminology of group actions.

    HW #6 will be due 11/9/21. But in addition to questions in the lecture summary text, start thinking about...
  • Does the map you gave in problem (3c) of HW #5 restrict to an isomorphism between Zmn× and Zm× × Zn×? In what circumstances?
  • Show that (a) the kernel of a homomoprhism is a subgroup of the domain; (b) the image of a homomorphism is a subgroup of the codomain; (c) a homomorphism is injective if and only if the kernel is trivial.
  • Try exercises 5.4.1 and 5.4.2, especially if cycle notation for permutation groups is new to you.
  • Th

    Orders of elements of Zm
    • For [a] in Zm we have ord([a]) = m if and only if gcd(am) = 1 (Corollary 4.14). Therefore Zm has φ(m) generators.
    • More generally, ord([a]) = m/gcd(am) (Theorem 4.13).
    • Questions to think about:
      • We saw that every element of Zm has order dividing m. For d a divisor of m, how many elements of Zm have order d?
      • We saw earlier that every subgroup of Zm is cyclic (Theorem 4.10). Why must every cyclic subgroup of Zm have order dividing m? For d a divisor of m, how many cyclic subgroups of order d are there in Zm?
      • What can you conclude about Σd|m φ(d)?
    Symmetries of a square: 8 elements (four rotations, four flips). Not cyclic. Not abelian. Generated by r, the rotation by 90, and f, the flip about the vertical axis. The element r2 is central: it commutes with all elements of Symm(□). Subgroup diagram: one cyclic subgroup of order 4, two Klein-4 subgroups (one is the symmetries of a fat vertical or fat horizontal line through the center, the other is the symmetries of a fat diagonal line), and five subgroups of order 2. Here the Klein-4 group is the noncyclic group of order 4, isomorphic to Z2 × Z2 and with presentation ⟨AB | A2 = B2 = 1, AB = BA⟩. Symm(□) is not isomorphic to the quaternion group Q8.

    Next time: Permutation groups!

    Every subgroup of a cyclic group is cyclic, again (Theorem 4.10).

    Corollary: Every subgroup of Z has the form aZ for some integer a, determined uniquely up to sign, so that the subgroups of Z are in one-to-one correspondence with nonnegative integers, with containment of subgroups corresponding to being a multiple of for integers (that is, aZ ⊆ bZ if and only if b | a). Finally, gcd(ab) is the nonnegative generator of the subgroup ⟨ab⟩ = bZ + bZ.

    Definition of kernel and image of a homomorphism of groups. Without proof: the image is a subgroup of the codomain; the kernel is a subgroup of the domain; a homomorphism is injective if and only the kernel is trivial. More on all this soon.

    Examples of homomorphisms.
    • exp : R → R+ is an isomorphism of groups
    • det : GL2(R) → R× is a surjective homomorphism.
    • The inclusion of any subgroup into the ambient group is an injective homomorphism.
    • Any group surjects onto the trivial group.
    • The reduction-modulo-m map is a surjective homomorphism Z → Zm.
    • The map φ : Z4 → Z2 × Z2 defined by φ([a]4) = ([a]2, [a]2) is a homomorphism, neither injective not surjective; both ker φ and im φ are isomorphic to Z2, and these consituent pieces “fit together” to make Z4, but in a way that's different from the way that two copies of Z2 fit together to make Z2 × Z2.

    Presentations, and maps out of groups defined by a presentation
    • A presentation for Symm(▵) in terms of generators and relations is ⟨rf | r3 = 1, f2 = 1, fr = r2f⟩. To give a homomorphism from Symm(▵) to a group G is to choose two elements gr and gf in G satisfying gr3 = 1, gf 2 = 1, and gf gr = gr2gf. Then the map sending r to gr and f to gf uniquely determines a well-defined homomorphism Symm(▵) → G.
    • A presentation for Z is ⟨ a |  ⟩ — no relations: Z is a free group. To give a homomorphism from Z to any group G is to pick out an element g of G; the map 1 ↦ g determines a unique homomorphism Z →G.
    • A presentation for Zm is ⟨ a | am = 1⟩. To give a homomorphism from Zm to a group G is to pick out an element g of G satisfying gm = 1, and then 1 ↦ g determines uniquely a well-defined (why??) homomorphism Zm →G.
    Midterm handout.

    Units modulo m
    • Bézout’s lemma lemma implies that Zm× = {[a]m : gcd(am) = 1}. See Example 3.11; note that Judson writes U(m) for Zm×.
    • The Euler phi function is the number of elements in Zm×:
      φ(m) := |{1 ≤ a ≤ m : gcd(am) = 1}|. This function is sometimes multiplicative, sometimes not: φ(3)φ(5) = φ(15) but φ(2)2 ≠ φ(4).
    Orders of group elements
    • The order of an element g of a group. This is sometimes denoted |g|, but we will write ord(g) for clarity.
    • Every element of a finite group has a finite order, and that order is at most the size of the group. The proof uses the Pigeonhole Principle.
    • Let g be an element of a group. Then for any integer n we have gn = 1 if and only if ord(g) | n (Proposition 4.12).
    Structure of cyclic groups
    • Suppose G is a cyclic group, generated by g ∈ G. If ord(g) is infinite, then G is isomorphic to Z. If ord(g) = m in Z+, then G is isomorphic to Zm. See also Theorem 9.7 and Theorem 9.8.
    • Corollary: An element of order m generates a group of order m.
    • Every subgroup of a cyclic group is cyclic (Theorem 4.10).

    No lecture (BU Monday)
    HW #5
    (due 10/19/21)
    LaTeX file

    A little more number theory: still section 2.2.
    • End of the proof of Bézout’s lemma (Theorem 2.10). For a, b nonzero, gcd(ab) is therefore also the least positive integer expressible as an (integer-) linear combination of a and b — that is, as ax + by for some x, y in Z.
    • We define gcd(0, b) := |b| for any b in Z, including for b = 0. This allows us to define gcd(ab) for any pair of integers a, b. [If one of a, b is nonzero, this definition agrees with the original one (it is still a positive common divisor of a and b that is a multiple of every other common divisor, and it's even still the least positive integer-linear combination of a and b. If a = b = 0, things get more fuzzy, and it’s best just to accept gcd(0, 0) := 0 as the definition.]
    • Euclid’s algorithm for computing d = gcd(ab). This algorithm is used to find x, y so that ax + by = d. (See second part of 2.2, called “Euclidean algorithm”.)
    • Defintion of prime, composite for positive integers. (Note that 1 is neither prime nor composite; it’s what’s called a unit because it has an integer inverse.)
    • Theorem: Every integer n > 1 has a factorization into primes. (See Existence part of the proof of Theorem 2.15.)
    • Theorem (Euclid, Theorem 2.14): There are infinitely many primes.
    • Fundamental lemma (Theorem 2.13): If a, b are integers, and p is a prime dividing ab, then p divides a or p divides b.
    • Unique factorization in Z (also known as the Fundamental theorem of arithmetic, Theorem 2.15): every integer n > 1 has a unique factorization into primes.
    In Z, we have a chain of reasoning: division algorithm ⇒ Bézout’s lemma ⇒ fundamental lemma ⇒ unique factorization.
    [This chain of reasoning also works in other number systems, such as the Gaussian integers.]

    Next time: orders of elements. Structure of cyclic groups. See 4.1.


    A little bit of number theory: sections 2.1 and 2.2.
    • Division algorithm (Theorem 2.9): for every a, b in Z with b ≥ 1 there exist unique q, r in Z with 0 ≤ r < b satisfying a = bq + r.
    • The proof (not given in class) requires the well-ordering principle (WOP, Principle 2.6): every nonempty subset of Z+ has a least element. WOP implies (and is in fact equivalent to) the principle of mathematical induction (PMI, Principle 2.1): if S is a subset of Z+ with 1 ∈ S and satisfying the property that n ∈ S implies n + 1 ∈ S for any positive integer n, then S = Z+.
    • Definition of a common divisor of two nonzero integers a, b.
    • The greatest common divisor (gcd) of a and b is defined to be a positive common divisor d of a and b with the property that any common divisor of a and b divides d. [Why does such an element exist? If it exists, why is it unique?]
    • Definition of what it means for a and b to be relatively prime.
    • Bézout’s lemma (Theorem 2.10): If a, b are nonzero in Z, then there exist x, y in Z so that ax + by = gcd(ab). So far we have shown that the minimal positive integer that can be expressed as ax + by is a common divisor of a and b.
    HW #4
    (due WED 10/13/21)
    (due 12pm THU 10/14/21)

    LaTeX file


    What we talk about when we talk about two groups having the “same Cayley table up to relabeling”: A map f : G → H between groups G and H is an isomorphism of groups if f (1) is a bijection and (2) preserves the group operation: that is, if for every a, b in G we have f(ab) = f(a)f(b). (More generally, f is a homomorphism of groups if f(ab) = f(a)f(b) for every a, b in G; and an isomorphism is a bijective homomorphism.) Isomorphisms (and more generally homomorphisms) map identity elements to identity elements and (exercise) inverses to inverses.

    Example: let H be the cyclic subgroup of Q generated by 2. Then the map Z → H sending n to 2n is an isomorphism. The homomorphism property is verified because 2m + n = 2m·2n for every integer m, n.

    Example: the map Z6 → Z2 × Z3 sending [x]6 to ([x]2, [x]3) is well-defined and an isomorphism of groups. Exercise: show that the map Z2 × Z3 → Z6 sending ([a]2, [b]3) to [3a + 4b]6 is well-defined, a group homomorphism, and the inverse to the first map.

    This is just a first taste of these ideas; we will return to them again and again. See also the first half of 9.1 and the beginning of 11.1.


    Some basic guidelines for how to write proofs. Please read Francis Su’s essay with some suggestions and examples.

    Subgroup diagrams for groups of order 1, 2, 3, 4, 5, 6. (This topic is indirectly covered in Judson in, for example, Example 4.7.)

    Cyclic subgroupa⟩ := {an : n ∈ Z} generated by an element a of a group G. This is the “smallest” subgroup of G containing a, in the sense that every subgroup of G containing a contains ⟨a⟩ (Theorem 4.3). If G = ⟨a⟩, then G itself is cyclic and a is a generator. (See section 4.1.) More generally for any subset S of G we define ⟨S⟩ as the smallest subgroup of G containing S; informally, this is the set of all “words” on elements of S and their inverses, evaluated in G, under concatenation.

    Symm(▵) is generated by the 120°-rotation r and the flip about the vertical axis f. The other rotation is r2 and the other flips are rf and fr.
    • Exercise (from class): Any two flips generate Symm(▵).
    • We know that r3 = f2 = 1. What other relations hold?
    • Are all the statements here still true if r replaced by either of the two rotations and f is replaced by any of the three flips?
    HW #3 redux
    (due 10/5/21)

    LaTeX file

    Equivalence relations, continued (section 1.2): An equivalence relation (ER) ∼ on a set S partitions S into equivalence classes [a] := {b ∈ S : b ∼ a}. The symmetric and transitive property guarantee that the equivalence classes are either disjoint or coincide, and the reflexive property guarantees that their union is all of S. The set of equivalence classes is denoted S/∼. Key example: we redefine Zm to be Z/(congruence modulo m), and call the equivalence classes congruence classes. Another example: let F := Z × Z – {0} be the set of all fractions, and then Q = F/∼, where a/b ∼ c/d if ad = bc.

    Maps from sets of equivalence classes: Let f : S → T be a map between sets, and ∼ an ER on S. If whenever a1 ∼ a2 in S we have f(a1) = f(a2), then f induces a well-defined map S/∼ → T given by [a] ↦ f(a). Similarly, if ⚬ : S × S → S is binary operation on S, and whenever a1 ∼ a2 and a1 ∼ a2 we have a1 ⚬ b1 ∼ a2 ⚬ b2, then ⚬ induces a well-defined binary operation on S/∼ given by [a] “⚬” [b] = [a ⚬ b]. Moreover if ⚬ is associative on S, then the induced binary operation on S/∼ is automatically associative as well. In this way we show that addition and multiplication is well-defined and associative on integers modulo m (section 3.1).

    For more on well-definedness of maps on a set of equivalence classes, see these very nice notes from Prof. Allan Yashinski.

    Subgroup diagrams for groups of order 1, 2, 3,...

    For practice, show that negation, addition, and multiplication are all well-defined on Q viewed as equivalence classes of fractions.

    Sirui’s example: Consider the subset {[4], [2]} of congruence classes modulo 6 under multiplication modulo 6. We have [4]2 = [4],
    [2][4] = [4][2] = [2], and [2]2 = [4]. Is this a group? A subgroup of Z6? A subgroup of Z6×?

    Subgroups (section 3.3): definitions, subgroup test (Proposition 3.31). Number system examples, both additive and multiplicative.

    The circle group and the mth roots of unity (second half of section 4.2): The unit circle T := {x ∈ C : |x| = 1} is a subgroup of C×. The T stands for torus; another common notation is S1, for the 1-sphere. For every positive integer m, the set T[m] := {x ∈ C : xm = 1} = {e2πik/m : k = 0, 1,..., m–1} of mth roots of unity is a subgroup of T of order m.

    Equivalence relations (last part of section 1.2): Relations, definition and examples. (Connection with functions.) Reflexive, symmetric, and transitive relations. Equivalence relations. Examples: equality, congruence modulo m. An equivalence relation on a set S defines a partition of S into equivalence classes and conversely (Theorem 1.25).

    HW #3
    (due 10/5/21)

    LaTeX file
    The invertible elements in various number systems forms groups under multiplication: R× := R ‑ {0}, Q× := Q ‑ {0}, C× := R ‑ {0}, and Z× := {1, ‑1}.

    Direct products of groups. See also problem (1) on HW #2.

    Brief review of 2×2 matrices. M2(R) as a group under addition: same group structure as R4 up to relabeling. Determinants and invertibility. GL2(R) and SL2(R) as groups under matrix multiplication (Examples 3.14, 3.26). [More generally, for any number system A, the set of 2×2 matrices with coefficients in A whose determinants are invertible in A form a group under multiplication.]

    For a, b elements of a group, (a‑1)‑1 = a and (ab)‑1 = b‑1 a‑1 (Propositions 3.19, 3.20). Notation: an for n in Z; usual rules of exponents hold, and we often write 1 for the identity element (Theorem 3.23). With certain abelian groups, it’s more appropriate to use additive notation: write a + b for ab and 0 for the identity element; be careful here -- it’s really easy to get confused!

    Multiplicative structure on Zm; note that we have not yet shown that multiplication is associative! The subset Zm× := {a in Zm : there exists b in Zm with a ×m b = 1} is a group under the operation “multiplication modulo m”. Examples for m = 2, 3, 4.

    Left/right cancelation (Proposition 3.22). Cayley table proposition: every element of a group G appears exactly once in each row and in each each column of the Cayley table for G. Equivalently, for every a, b in G, the equation ax = b has a unique solution and the equation xa = b has a unique solution.
    • Proof 1, for finite groups: consider the left-multiplication-by-a map ℓa: GG. Left cancelation tells us that ℓa is injective. Since the domain and the codomain are both finite sets of the same cardinality, ℓa is surjective as well. This proves that each row of the Cayley table is a permutation of the elements of G. Do the same for the right-multiplication-by-a map and columns.
    • Proof 2: see Proposition 3.21.
    Up to relabeling there is exactly one Cayley table for a group of order 3. For example, the group of rotations of an equilateral triangle has this Cayley table.

    Integers modulo m (Example 3.9): For every integer m ≥ 1, the set {0, 1,..., m–1} under the operation “addition modulo m” forms an abelian group of order m, denoted Zm. We will give a more elegant construction for this group very soon.

    Groups of order 4: integers modulo 4, the group of symmetries of a rectangle...

    Still 3.2

    HW #2
    (due 9/21/21)
    Cartesian products of sets. A binary operation on a set S is a map S × S → S.

    In a group, the identity element is unique (Proposition 3.17). More generally, if a set with a binary operation has an identity element, this identity is unique.

    In a group, inverses are unique (Proposition 3.18). More generally, in a set with an associative binary operation with an identity element, if an element has both a left inverse and a right inverse, then these are equal and the element has a (unique, two-sided) inverse. [A set with an associative binary operation that has an identity element is called a monoid. Examples: Z≥0 under addition; Z under multiplication.]

    Examples of groups: Symm(▵), Symm(C/R) = {id, complex conjugation}, (Z, +), (Q, +), (R, +), (C, +), (2Z, +), ({0}, +).

    If aa = a in a group (G, ⚬), then a is the identity element of the group.

    Cayley tables (group tables) for groups of order 1 and groups of order 2.

    More 3.2

    Z, Q, R:
    see 1.2

    see 4.2
    Challenge: finish finding all possible Cayley tables for groups of order 3 up to relabeling!
    Functions: domain, codomain, fS → T notation, image f(A) ⊆ T for a subset A of S, preimage f –1(B) ⊆ S for a subset B of T.

    Injective (one-to-one) functions; a function f: ST is injective if and only if for every tT we have | f –1({t})| ≤ 1; connection with horizontal line test. Surjective (onto) functions; a function f: ST is injective if and only if for every tT we have | f –1({t})| ≥ 1. Bijective functions. Examples.

    Compostion of functions: notation, composition of functions is always associative (Theorem 1.15). Identity function idS : S → S and its properties. Inverse functions. Two-sided inverses are unique. A function has an inverse if and only if it is bijective (Theorem 1.20).

    If S and T are two finite sets and f: ST is a function, then always |S| ≥ |f(S)| ≤ |T|. Claim: f is injective ⇔ |S| = | f(S)|; f is surjective ⇔ | f(S)| = |T|. Corollary: If S and T are two finite sets of the same cardinality and f: ST is a function, then f is injective ⇔ f is surjective ⇔ f is bijective.

    1.1, 1.2
    (through Theorem 1.20)
    Introduction; syllabus review.

    Symmetries of an equilateral triangle. Definition of a group. First examples.
    (edit 9/9/21: section number typo corrected)
    HW #1
    (due 9/14/21)
    (edit 9/9/21:
    Ok to turn in 9/16/21)

    (edit 9/14/21: email address typo corrected)

    Additional resources