MODERN ALGEBRA (MATH 210) PROBLEM SET 2

From the text: 2.69, 2.76, 2.98, 2.103. (For the last, and for future reference, read and understand the proof of Burnside’s Lemma, Theorem 2.113. Caution: the answer given in 2.103(ii) is incorrect!)

A1. This exercise shows that for n ≠ 6, every automorphism of S_{n} is inner. Fix an integer n ≥ 2 with n ≠ 6.

(a) Prove that the automorphism group of a group G permutes the conjugacy classes of G, i.e. for each σ ∈ Aut(G) and each conjugacy class K of G the set σ(K) is also a conjugacy class of G.

(b) Let K be the conjugacy class of transpositions in S_{n} and let K′ be the conjugacy class of any element of order 2 in S_{n} that is not a transposition. Prove that |K| ≠ |K′|. Deduce that any automorphism of S_{n} sends transpositions to transpositions.

(c) Prove that for each σ ∈ Aut(S_{n})

σ : (12) ↦ (ab_{2}), σ : (13) ↦ (ab_{3}), …, σ : (1n) ↦ (ab_{n})

for some distinct integers a, b_{2}, b_{3}, …, b_{n} ∈ {1, 2, …, n}.

(d) Show that (12),(13), …, (1n) generate S_{n} and deduce that any automorphism of S_{n} is uniquely determined by its action on these elements. Use (c) to show that S_{n} has at most n! automorphisms and conclude that Aut(S_{n}) = Inn(S_{n}) for n ≠ 6.

A2. We now show that Inn(S_{6}) is of index at most 2 in Aut(S_{6}). Let K be the conjugacy class of transpositions in S_{6} and let K′ be the conjugacy class of any element of order 2 in S_{6} that is not a transposition. Prove that |K| ≠ |K′| unless K′ is the conjugacy class of products of three disjoint transpositions. Deduce that Aut(S_{6}) has a subgroup of index at most 2 which sends transpositions to transpositions. Then prove that | Aut(S_{6}) : Inn(S_{6})| ≤ 2.

A3. Finally, we exhibit an outer automorphism of S_{6}. (There are other, more beautiful, descriptions.) Let t′_{1} = (12)(34)(56), t′_{2} = (14)(25)(36), t′_{3} = (13)(24)(56), t′_{4} = (12)(36)(45), t′_{5} = (14)(23)(56). Show that t′_{1}, …, t′_{5} satisfy the following relations:

• (t′_{i})^{2} = e for all i;

• (t′_{i}t′_{j})^{2} = e for all i and j with |i − j| ≥ 2;

• (t′_{i}t′_{j})^{3} = e for all i and j with |i − j| = 1.

Use this to show that the map (i(i+1)) ↦ t′_{i} gives an automorphism of S_{6}. (In the process, you will likely have to show that the relations above define S_{6}. Your argument will also presumably prove the obvious generalization to S_{n}.)

Date: Tuesday, October 15, 2002.

B1. If there exists a chain of subgroups G_{1} ≤ G_{2} ≤ ⋯ ≤ G such that G = ∪_{i=1}^{∞}G_{i} and each G_{i} is simple, then G is simple. (Note that G need not be finite!)

B2. (a) Let Ω be an infinite set. Let D the subgroup of S_{Ω} consisting of permutations which move only a finite number of elements of Ω and let A be the set of all elements σ ∈ D such that σ acts as an even permutation on the (finite) set of points it moves. Prove that A is an infinite simple group.

(b) Prove that if H ≠ {e} is a normal subgroup of S_{Ω}, then H contains A, i.e. A is the unique nontrivial minimal normal subgroup of S_{Ω}.

C. For any finite group P, let d(P) be the minimum number of generators of P (so, for example, d(P) = 1 iff P is a nontrivial cyclic group). Let m(P) be the maximum of the integers d(A) as A runs over all abelian subgroups of P. Define

J(P) = ⟨A : A is an abelian subgroup of P with d(A) = m(P) ⟩.

(J(P) is called the Thompson subgroup of P. It plays a pivotal role in the study of finite groups, and in particular the classification of finite simple groups.)

(a) Prove that J(P) is preserved by all automorphisms of P. (This is the definition of a characteristic subgroup.) Hence show that J(P) is normal.

(b) For both P = Q_{8} and D_{8}, list all abelian subgroups A of P that satisfy d(A) = m(P). In both cases show that J(P) = P.

(c) Prove that if Q ≤ P and J(P) is a subgroup of Q, then J(P) = J(Q). Deduce that if P is a subgroup (not necessarily normal) of the finite group G and J(P) is contained in some subgroup Q of P such that Q is normal in G, then J(P) is normal in G as well.

Problems A1–C are from Dummit and Foote.

The set is due Tuesday, October 22 at 3:30 pm in Pierre Albin’s mailbox (opposite the elevator on the first floor of Building 380).