Course webpage for MA 741, Algebra I, Fall 2020

Instructor: Anna Medvedovsky

Class meetings and homework assignments

Date Lecture Topics Homework
Tu 12/10/2020
Lecture 27
Last class: Long exact sequence attached to a left derived functor (long exact seqeunce in homology given a short exact sequence of complexes; Horseshoe lemma). Tor1(Q/Z, A) and the torsion group of A. Categorical limits and colimits. Dual group of Q/Z. Course evaluations.

For the curious, a clarification/correction to what I said in class about guaranteeing that a functor has an adjoint: the Adjoint Functor theorem describes the conditions that a functor preserving (co)limits has to satisfy in order to have a (right)left adjoint. See section 4.3 in Pavel Safronov's course notes or Kevin Buzzard's notes thereon; latter is always a fun read.

Winter reading: Ext groups and extensions.
• Compute Ext1Z(Z/nZ, M) for a Z-module M by hand: take a free resolution of Z/nZ, apply HomZ(—, M), and take cohomology.
• See Aluffi exercises VIII.6.21-22 for the connection between Ext1R(N, M) and R-modules E that are extensions of M by N: that is, that fit into short exact sequences 0 → MEN → 0. Work through this connection by comparing Ext1Z(Z/nZ, Z/mZ) to what you get from the structure theorem for finite abelian groups, say, for prime power m and n.
• Let G = Z and compute Ext1C[G](C, χα), where χα: GC* is a character sending a generator to α in C* (see Aluffi Example IX.7.14 for a similar computation). Then relate this to HW #7 problem 4(c) via problems 5(a)&(d). What about Ext1C[G]β, χα)?
Dual story of derived functors: given a left-exact additive functor such as HomR(N, —) get a family of right derived functors measuring the failure of exactness of the original functor, by dually taking cohomology of the complex obtained after applying the original functor to an injective resolution of the object in question. (Here obviously an R-module Q is injective if HomR(—, Q) is exact, what else could it possibly be?) And then you're ready to go wild! Group cohomology! Add in some Galois theory (Hilbert 90, Kummer theory, Brauer groups,...)! Sheaves and sheaf cohomology! Or for a different direction, the profinite topology on Zp and other abelian profinite groups, Pontryagin duality between Q/Z and Z, profinite groups more generally, infinite Galois theory...
Take good care.
Tu 12/10/2020
Lecture 26
(Co)chain complexes, (co)homology. Definition of TornA(M, N) again via a free resolution of M. Computation via free resolution of Z/mZ: Torn(Z/mZ, N): Tor0(Z/mZ, N) = N/mN = Z/mZAN; Tor1(Z/mZ, N) = N[m], the m-torsion (hence the name); and the higher Tor groups vanish. More generally, Tor0A(M, N) = MAN. Computation: when M is free (at least free of finite rank), TornA(M, N) vanishes for n ≥ 1 for every N. Similarly, if N is flat (for example, free) then TornA(M, N) vanishes for n ≥ 1 for every M; this is a characterization of flat modules (unproved) (and Tor1A(M, N) vanishing for every M is enough). In fact, because tensor product is commutative, so are the Tor functors (unproved), so these statements in the last two sentences are not unrelated.

Torn(—, N) is a family of well-defined functors: Morphisms of complexes induce maps between homology. Projective modules: P is projective A-module if HomA(P, —) is exact: that is, if maps from P lift along surjections. Free modules are projective. If f : MN is a map of A-modules, and FM and GN are projective (for example, free) resolutions then f lifts to a morphism of complexes FG (DF 17.1 Prop. 4), and moreover the induced maps on homology depend only on f, not on the lift to FG (DF 17.1 Prop. 5; see Weibel 1.4 for more on chain homotopies). As a corollary, the Torn(M, N) groups as defined here do not depend on the particular free, or even projective, resolution of M chosen (DF 17.1 Theorem 6); and moreover Torn(—, N) are functors. Furthermore the same construction works with any additive right-exact functor F: R-modS-mod: we get a family of left derived functors LnF: R-modS-mod with L0F naturally isomorphic to F.

Long exact sequence of dervied functors: statement. Example: If 0 →MNP → 0 is an exact sequence of abelian groups, then the long exact sequence derived from applying the functor — ⊗Z Z/mZ is 0 →M[m] → N[m] → P[m] → M/mMN/mNP/mP → 0.

For the curious: see a short exact sequence of complexes leading to the long exact sequence in homology proved with the Snake lemma in Weibel pp.13-14.
Nevermore!
Tu 12/3/2020
Lecture 25
Proof of the primitive element theorem. Side proposition: a finite field extension L/K has ≤ [L:K] embeddings to any algebraic closure , with equality if and only if L/K is separable. Proof of the Galois correspondence for finite Galois extensions. Example: Galois correspondence for Q(√ 2, √ 3) over Q.

Finite fields: There is a field of order pn for any prime p and any n: take the roots of xpn - x in p (that's 4(c) on HW #8). Conversely, any field of order pn is a splitting field of the polynomial xpn - x. Corollary: all finite fields of the same size are isomorphic. Example: Gal(Fqd / Fq) ≅ Z/dZ for any q = pn. (What are the intermediate fields and how do they correspond to intermediate subgroups?)

A teaser of derived functors: desired properties for left derived functors of an additive right exact functor from A-mod to A-mod, construction of Torn(M, N) for A-modules M, N via a free resolution of M.

References: Field theory and the Galois correspondence: DF 13.1-2, 13.4-5, 14.1-4; Aluffi VII.1, VII.2.1, VII.4-6. A bit of homological algebra: DF 17.1.
Homework assignment #9
(Hint for 3a edited 12/14/2020.)
Tu 12/1/2020
Lecture 24
Normal extensions and splitting fields. Detailed study of the extension Fp(t)[x]/(xn - t) over Fp(t) and its automorphisms for n = 2, 3. Separable extensions. Galois (that is, normal and separable) extensions. Primitive element theorem: every finite separable extension is simple (statement only; see DF Theorem 25 on p.595 or Aluffi VII.5.19 or comment below for proof). Corollary: if L/K is finite Galois, then the cardinality of the group AutK(L), also known as the Galois group Gal(L/K), is equal to [L:K]. Statment of the fundamental theorem of Galois theory: if L/K is finite Galois, then there is an inclusion-reversing correspondence between intermediate subfields and subgroups of Gal(L/K) sending a subextension E/K to Gal(L/E) and a subgroup H to the subfield LH of L fixed by H. Moreover, subfields E of L normal over K correspond to normal subgroups of Gal(L/K). [Not stated in class: in this normal case, Gal(E/K) is isomorphic to the quotient Gal(L/K) / Gal(L/E).]

After-hours example: the Galois correspondence for Q(21/3, ω) over Q.

(Both book references above deduce the primitive element theorem from a theorem of Artin that says that a finite extension is simple if and only if there are only finitely many intermediate extensions. Artin's theorem allows you to prove that extension such as Fp(t1/p, u1/p) over Fp(t, u) are not simple. But if you're just interested in the primitive element theorem itself, the quick argument I had in mind is on Wikipedia.)
Work out the Galois correspondence for Q(21/3, ω) over Q if you've never done it, or not recently.
Tu 11/24/2020
Lecture 23
More field theory. Set of algebraic elements in an extension is a subfield. Algebraic extensions of algebraic extensions are algebraic (DF 13.2 Theorem 20, Aluffi Corollary VII.1.18) Algebraically closed fields. Algebraic closure of a field. Existence of algebraic closure (DF 13.4 Propostion 30, Aluffi section VII.2.1). Uniquess of algebraic closure up to isomorphism (Aluffi p.404). Example: Q(21/3) over Q redux, Q(ω) over Q where ω is a primitive cube root of 1, Q(21/3, ω) over Q(21/3), and Q(21/3, ω) over Q(ω). Example: F = F3(α) over F3 where α satisfies x2 + 2x + 2: the map aa3 (the Frobenius automorphism) is an order-2 automorphism of F fixing F3, and therefore x2 + 2x + 2 = (x - α)(x - α3) in F[x].
Homework assignment #8
(Typos corrected 12/11/2020.)
Th 11/19/2020
Lecture 22
Detour to field theory. Field extensions, degree of an extension. Simple extensions. Elements algebraic over the base field and algebraic extensions; transcendental extensions. Simple extensions are either finite or transcendental. Primitive element of a simple extension; minimal polynomial of an element; every polynomial satisfied by an element is a multiple of the minimal polynomial. Finitely generated extensions. A finitely generated extension is finite iff algebraic iff generators are algebraic. Degrees of extensions are multiplicative in towers (see DF Theorem 13.14 or Aluffi Prop. V.1.10). Automorphisms of a field extension. The number of automorphisms of a finite simple extension is equal to the number of roots of the minimal polynomial of a primitive element in the extension, which is less than or equal to the degree of the extension. Example: Q(i) over Q: degree-2 simple extension, automorphism group Z/2Z. Example: Q(21/3) over Q: degree-3 simple extension, trivial automorphsim group (minimal polynomial of 21/3 doesn't split in Q(21/3)!).
What are the automorphsims of Fp(t)[x]/(xn - t) over Fp(t)? The answer will depend on both p and n! In particular, think about the case n = 2 and 3 for various p as well as the case n = p.
Tu 11/17/2020
Lecture 21
Tensor products of algebras. Coproduct in the category of commutative A-algebras and the category of rings. Example: CRC. Family of examples: Z[x]/(x2 + 1) ⊗Z Fp: these are Fp-algebras whose shape (field, product of fields, ring with nilpotents) keeps track of the way x2 + 1 factors modulo p. For a commutative ring A and A-modules M, N, the abelian group HomA(M, N) is an A-module in its own right. Hom-tensor adjunction again: HomA(N, —) is right adjoint to — ⊗A N. The functor — ⊗A N is right exact. More generally, left adjoints are right exact and right adjoints are left exact (not proved in detail). Flat modules. Localization is flat (not proved). Free modules are flat.
Th 11/12/2020
Lecture 20

More on tensor products of modules over a commuative ring. Definition via UP. Existence. Properties. Mention of the hom-tensor adjunction: more on it soon. — ⊗A N as a functor. Algebras over a commutative ring: an A-algebra structure on a ring B is an A-module structure so that muliplication is A-bilinear: in other words, a ring homomorphism AB landing in the center of B. [Not mentioned in class: these are unital associative algebras (in contrast, to, for example, Lie algebras over a field).] Restriction and extension of scalars as adjoint functors between A-mod and B-mod for an A-algebra B. Examples: QZ G for G a finite (or finitely generated) abelian group. Complexification of a real vector space as a tensoring operation. Example: V = R2 with R-linear operator T: VV with T((1,0)) = (0,1) and T((0,1)) = (-1,0). To find a basis for V in which T looks diagonal, need to pass to VRC.
Tu 11/10/2020
Lecture 19

More on R-modules and the category R-mod. Ab-categories (categories where Hom-sets are abelian groups, and composition distributes over this addition). Examples: Ab and R-mod. In an Ab-category C the endomorphism group EndC(A) has a ring structure for every A in C; a (left) R-module is an abelian group M together with an action of R on M in the form of a ring homomorphism R → EndAb(M). Additive functors between Ab-categories (maps on Homs are homomorphisms of abelian groups). Examples: for an abelian group X, the functors hX = HomAb(X, —) and hX = HomAb(—, X) are additive functors from Ab to itself. Zero objects (objects that are both final and initial). Examples: trivial group in Group, zero group/rng in Ab or Rng. Given a category and a zero object, any Hom-set has a distinguished "zero" map passing through the zero object. In an Ab-category with a zero object, this zero map is the additive identity of each Hom-group (exercise). Products and coproducts in the category of R-modules. In a category with zero objects, coproducts map nicely to products. In any Ab-category with a zero object and finite products, finite products are also coproducts (exercise). Free R-modules (linearly independent sets, spanning/generating sets, bases). UP satisfies by free R module FR(S) on a set S. Again free module construction is left adjoint to forgetful functor SetR-mod. Tensor products of modules over a commutative ring: construction. Example Z/2ZZ Z/3Z = 0.
References. Tensor products: Atiyah-Macdonald Chapter 2; DF 10.4 (but DF does not assume that the underlying ring is commutative!); Aluffi VIII.2. Free modules and products/coproducts: DF 10.3; Aluffi III.6.2, III.6.1. Ab-categories: Weibel A.4.
Th 11/5/2020
Lecture 18
(only half!)

Prof. Robert Pollack lectures on modules. Definitions, submodules, quotient modules, homomorphisms, UP for quotients, submodule corresponence for quotients, sums and intersections of submodules, isomorphism theorems, direct sums and direct products of modules. Structure theorem for finitely generated modules over a PID. Corollary: a finitely generated torsion-free module over a PID is free. Non-example: the ideal (2, 1 + √-3) of the ring Z[√-3] is finitely generated and torsion free but not free. Modules over K[x] ↔ K-vector spaces with a K-linear operator (see also DF example p.340 or Aluffi VI.7.1). Jordan canonical form for matrices over K = C as a corollary of the structure theorem.

References: DF 10.1-3, 12.1&3. Aluffi III.5, VI.7.1&3.
At the end of problem 6 of HW #5 you showed that Z[(1 + √-3)/2] is Euclidean, and hence a PID. What happens to the ideal (2, √-3) in that ring?
Tu 11/3/2020
Lecture 17
Presheaf of sets (or abelian groups or commutative rings) on a topological space T as a contravariant functor from Open(T) to Set (or Ab or CRing).

Natural transformations. Definitions: natural transformation, natural isomorphism. Examples: for a group G the homomorphism G → Gab is natural in the sense that there is a natural transformation from IdGroup to (—)ab as functors Group → Group. For a real vector space V, the linear map from V to its double dual V** taking a vector v to the evaluation-at-v map is natural in the sense that there is a natural transformation IdR-Vect ⟿ (—)** as functors from the category of real vector spaces R-Vect to iself, which restricts to a natural isomorphism on the subcategory R-fdVect of finite-dimensional real vector spaces.

Equivalences of categories. Definitions: full functor, faithful functor, essentially surjective functor, equivalence of categories. Equivalence of categories between R-Vect and category C with Ob(C) = Z≥ 0 and HomC(n, m) = Mn×m(R).

Adjoint functors. "Free" constructions as left adjoint to forgetful functors. Examples: For a set S and group G we have HomSet(S, Forget(G)) = HomGroup(F(S), G), capturing the universal property of free groups. For a set S and an abelian group A we have HomSet(S, Forget(A)) = HomAb(Fab(S), A), capturing the universal property of free abelian groups. The adjunction of abelianization functor Group → Ab and the inclusion functor Ab → Group captures the universal property of abelianization.

References: Aluffi VII.1.3, VII.1.5; DF Appendix II; Weibel Appendix A. For more on equivalences of categories, see, for example, section 1.7 in Pavel Safronov's course notes.
Let FinAb be the category of finite abelian groups. Show that the association of A in FinAb to A^ := HomAb(A, Q/Z) is a contravariant (dualizing) functor from FinAb to itself. Show that the isomorphism from 6e of HW #6 is a natural isomorphism IdFinAb ⟿ (—)^^ from the identity functor to the double-dual functor.
Th 10/29/2020
Lecture 16
Definitions: small category (class of objects is a set), concrete category (objects are sets, maps are set maps: in other words, faithful forgetful functor to Set), isomorphism (arrow with two-sided inverse), automorphism group AutC(A) (set of isomorphisms AA in C); groupoid (category where every arrow is invertible). Example: fundamental groupoid of a topological space. Examples of small categories: a monoid, a group, a set with a reflexive and transitive relation, integers under ≤, power set of a set under ⊆. Opposite category. Universal properties: initial and final objects; products and coproducts, categorically and as final/initial objects in categories with additional structure map data (so-called comma categories). Functors. Covariant functors: forgetful functor, abelianization, units of rings. Contravariant functors: Spec of a commutative ring. Covariant functor HomC(X, –) from category C to Set. Contravariant functor HomC(–, X) from category C to Set.

References: Aluffi I.4, I.5, VII.1.1, VII.1.2; DF Appendix II; Weibel Appendix A.
Check that abelianizing a group, passing to the unit group of a ring, taking Spec of a commutative ring, HomC(–, X), and HomC(X, –) are functors, covariant or contravariant as claimed. For abelianization, is there more to show than what's in problem 5 on HW #2? Same question for Spec-taking and problems 5 on HW #5 and 4 on HW #6.
Tu 10/27/2020
Lecture 15
Chinese remainder theorem (AM Prop 1.10, DF 7.6, Aluffi V.6.1). Understanding R[x]/(x2-d) for various real d.

Language of category theory. Definition of a category. Subcategories and fullness. Examples: Set of sets, Group of groups, Ab of abelian groups, Rng of rngs, Ring of rings, CRing of commutative rings, A-CAlg of commutative A-algebras, G for a group G (one object whose endomorphism are G). References: Aluffi I.3, DF Appendix II, Weibel Appendix A.
Homework assignment #6
(Due 11/4/2020)

Also: Show that every (left) ideal of a product ∏i Ri of rings is of the form ∏i Ii for Ii a (left) ideal of Ri.
Th 10/22/2020
Lecture 14
Finishing the PID ⇒ UFD proof: in noetherian rings (more generally, in rings that satisfy the ACC for principal ideals), elements can be factored into irreducibles (Aluffi Prop. V.1.1). A domain is a UFD if and only if ACC for principal ideals holds and every irreducible element is prime (for a proof, see Aluffi Theorem V.2.5).

Field of fractions of a domain (Aluffi V.4.2); see also DF II.7.5 or AM chapter 3 for the more general construction of inverting elements of a multiplicatively closed subset S of a commutative ring A.

Gauss's lemma. If A is a PID, then A[x] is UFD. Therefore Z[x] is a UFD. More generally, it's true that A is a UFD if and only if A[x] is UFD; see Aluffi V.4.3 or DF II.9.3.

Zorn's lemma (Aluffi V.3.1, or otherwise DF Appendix I.2). Every nonzero ring has at least one maximal ideal (Aluffi V.3.2, AM Theorem 1.3, DF Prop. 11 in II.7.4); can require this maximal ideal to contain any fixed ideal. Every nonzero ring has at least one minimal prime ideal; can require this prime to be contained in any fixed prime ideal.

Next: Chinese remainder theorem. Then: the language of category theory (Aluffi I.3, I.4, I.5 (pleasant to read), Weibel Appendix A (efficient), or DF Appendix II (functional, many examples)).
Check that the general S-1A construction works as advertised!

If p1p2 ⊇ ... is a descending chain of prime ideals, check that their intersection is also prime. Note that an intersection of prime ideals is NOT in general prime!
Tu 10/20/2020
Lecture 13
Defintions: divisibility, associates, prime elements, irreducible elements. Nonzero prime elements are irreducible. In a PID, irreducible elements are prime. In a PID, nonzero prime ideals are maximal. Examples of extensions of Fp. Krull dimension.

Unique factorization domains (UFDs). Example of two fundamentally different factorizations into irreducibles: in Z[√-5], we have 2*3 = (1 + √-5)(1-√-5). In a PID, factorizations into irreducibles are unique because irreducibles are prime; still left to show that in any noetherian ring, and in particular in a PID, factorizations into irreducibles exist.

A commutative ring is noetherian ⇔ ascending chain condition (ACC) is satisfied for ideals ⇔ nonempty collections of ideals have maximal elements. (See Prop. 1.1 in Aluffi V.1.1: take M=R and then submodules of M are ideals of R.)
Th 10/15/2020
Lecture 12
More commutative algebra. If K is a field, then K[x] is a PID. Briefly: definition of a Euclidean domain; EDs are always PIDs. Commutative ring Venn diagram (see also Aluffi p. 243): noetherian rings, integral domains, PIDs, Euclidean domains, fields. Characteristic of a ring, of a field. Fields are commutative rings with no nonzero proper ideals. Any nonzero ring homomorphsim from a field is injective. Prime subfields: Q and Fp. Prime and maximal ideals. Examples from Z, Z[x]. An ideal is prime iff quotient by it is a domain. An ideal is maximal iff the quotient by it is a field. Maximal ideals are prime. A finite-index ideal is prime if and only if it is maximal.

Next time: In a PID, any nonzero prime ideal is maximal. Prime and irreducible elements. Noetherian rings and chain conditions for ideals. PIDs are UFDs. Zorn's lemma (see Aluffi V.3 for a very nice discussion). Maximal ideals exist.

References for these topics: DF 7.4, 8.1, 8.2, 8.3, 9.1, 9.2, 9.3. Atiyah-Macdonald Chapter 1. Aluffi III.4, V.1, V.2, V.3.

For more about Z[(1 + √-19)/2], the classic example of a non-Euclidean PID, see DF pp. 277, 282; Aluffi exercises V.2.18, V.2.21; and/or Theorem 5.12 in Keith Conrad's notes on Euclidean domains.

Left/right polynomial division works fine over general rings: see, for example, Aluffi p. 147. However, in the commutative setting there's a chain of reasoning that leads from division algorthm to unique factorization. This chain of reasoning fails over noncommutative rings (see Qiaochu Yuan's answer).
Homework assignment #5
(Clean copy)
Th 10/8/2020
Lecture 11
All rings commutative today! Generation of ideals. Noetherian rings. PIDs. Z is a PID. Polynomial algebras again. Universal property for polynomial algebras. Polynomials in A[x] vs. polynomial functions on A. Division algorithm in Q[x]. Q[x] is a PID. Division by a monic in A[x]. Isomorphism between A[x]/(x2 - d) and A[√d]. More generally, if f in A[x] is monic of degree n, then A[x]/(f) puts a ring structure on An.
What are the units and zero divisors of A[x] for a general commutative ring A? Show that polynomials over an infinite field are distinguishable by their polynomial functions.
Tu 10/6/2020
Lecture 10
More ring definitions. Units and zero divisors. In a finite ring every element is either a unit or a zero divisor. Corollary: finite integral domains are fields. UP for quotients of rings. Isomorphism theorem for rings. Ideal correspondence for quotient rings. Operations on ideals. Principal ideals.

Next time: some commutative algebra — polynomial rings, prime and maximal ideals. After: modules, the language of category theory.
Th 10/1/2020
Lecture 9
Problem 6 on HW #1. More ring theory: division rings, fields, units, center of a ring. Examples: Z[√d] (commutative ring), Q(√d) (field), Hamiltonians (division ring). Ideals and quotient rings.
Tu 9/29/2020
Lecture 8
Presentations. Free products of groups. Problem 5b on HW #1. Rings: definitions and examples. Note that our rings will always have a multiplicative identity. (More precisely, we say "ring" for what DF calls "ring with identity," and "rng" for what DF calls "ring.")

Start looking at DF pp.25‑28, sections 6.3, 7.1, 7.2.
Th 9/24/2020
Lecture 7
Smith normal form theorem: proof by a few examples and discussion.
Free groups: generating set, rank, universal property. Informal introduction of Cayley graphs. Cayley graph of the free group on two generators. Nelson-Schreier theorem: a subgroup of a free group is free (not proved in class; for an elegant argument using group actions on graphs, see J-P. Serre, Trees).

There are many online references for Smith normal form, but here's one whose presentation is similar to what happened in class. (Let R = Z; for the norm function N(r) in Euclid's algorithm take the absolute value of r).
T 9/22/2020
Lecture 6
Recognition theorem for direct products of groups. Short exact sequences of groups. Splitting short exact sequences of abelian groups. A surjection onto a free group always splits. Any subgroup of a rank-n free abelian group is free abelian of rank ≤ n. Smith normal form theorem implies the structure theorem for finitely generated abelian groups.

Next time: sketch of proof of the Smith normal form theorem. Afterwards: free groups and presentations; semidirect products (?). Then rings and fields and modules.
Preliminary problems
The next homework set will be posted Thursday evening.
Th 9/17/2020
Lecture 5
Rank of a free abelian group is well defined. Universal property of free abelian groups. Statement of the fundamental structure theorem of finitely general abelian groups (Theorems 3 and 5 of DF 5.2), and the correspondence between the two statements. Towards a proof: an f.g. abelian group is the cokernel of an injective map between two f.g. free abelian groups, which we can express as a matrix. The goal is to find good bases for the two free abelian groups so that the matrix looks diagonal of the right shape.

Next time: how to find these good bases and Smith normal form. See exercise 19 (or even 16-19) in DF section 12.1 for a preview. Our R = Z, of course.
Start thinking about DF section 5.2 exercses 2, 3, 13, 14
T 9/15/2020
Lecture 4
Abelian groups: bases, freeness, torsion elements. Examples. Direct sums vs. direct products of abelian groups in terms of their universal mapping properties.
Homework assignment #2

Please include some recognizable part of your name, the number of the problem set, and (separately!) the number 741 in the document name of your homework submission.
Th 9/10/2020
Lecture 3
Group actions: Definitions, lots of vocabulary. So-called fundamental theorem of group actions (orbits partition; stabilizers are subgroups, and stabilizers of elements in the same orbit are conjugate; bijection between orbit and cosets of stabilizer). Examples. Action of finite p-group: cardinality of set is congruent to cardinality of set of fixed points modulo p. Center of a p-group is nontrivial. Groups of order p and p2 are abelian. In finite group G, any subgroup with the property that its index is the least prime dividing |G| is normal. See Dummit and Foote sections 1.7, 2.2, 4.1, 4.2, 4.3.