ABSTRACT ALGEBRA: OnLine Study Guide, Section 3.6

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§ 3.6 Permutation Groups

Definition 3.6.1. Any subgroup of the symmetric group Sym(S) on a set S is called a permutation group or group of permutations.

Theorem 3.6.2. (Cayley) Every group is isomorphic to a permutation group.

Definition 3.6.3. Let n > 2 be an integer. The group of rigid motions of a regular n-gon is called the nth dihedral group, denoted by D_n.

We can describe the nth dihedral group as

D_n= { a^k, a^kb | 0 k < n },

subject to the relations o(a) = n, o(b) = 2, and ba = a^-1b.

Proposition 3.6.4. The set of all even permutations of S_n is a subgroup of S_n.

Definition 3.6.5. The set of all even permutations of S_n is called the alternating group on n elements, and will be denoted by A_n.

Let _n be the polynomial in n variables x₁,x₂,...,x_n defined to be the product of all factors (x_i - x_j) in which 1 i < j n. Any permutation in S_n acts on _n by permuting the subscripts.

Lemma 3.6.6. Any transposition in S_n changes the sign of _n.

Theorem 3.6.7. A permutation in S_n is even if and only if it leaves the sign of _n unchanged.

§ 3.6 Permutation Groups: Solved problems

As in the previous section, this section revisits the roots of group theory that we began to study in Chapter 2. Cayley's theorem shows that permutation groups contain all of the information about finite groups, since every finite group of order n is isomorphic to a subgroup of the symmetric group S_n. That isn't as impressive as it sounds at first, because as n gets larger and larger, the subgroups of order n just get lost inside the larger symmetric group, which has order n!. This does imply, however, that from the algebraists point of view the abstract definition of a group is really no more general than the concrete definition of a permutation group. The abstract definition of a group is useful simply because it can be more easily applied to a wide variety of situation.

You should make every effort to get to know the dihedral groups D_n. They have a concrete representation, in terms of the rigid motions of an n-gon, but can also be described more abstractly in terms of two generators a (of order n) and b (of order 2) which satisfy the relation ba = a^-1b. In the problems in this section, we will use the following description of the dihedral group D_n of order 2n.

D_n = { aⁱ b^j | 0 i < n, 0 j < 2, o(a) = n, o(b) = 2, ba = a^-1b }

22. In the dihedral group D_n, show that baⁱ = a^n-i b, for all 0 i < n. Solution

23. In the dihedral group D_n, show that each element of the form aⁱ b has order 2. Solution

24. In S₄, find the subgroup H generated by (1,2,3) and (1,2). Solution

25. For the subgroup H of S₄ defined in Problem 24, find the corresponding subgroup H ^-1, for = (1,4). Solution

26. Show that each element in A₄ can be written as a product of 3-cycles. Solution

27. In the dihedral group D_n, find the centralizer of a. Solution

28. Find the centralizer of (1,2,3) in S₃, in S₄, and in A₄. Solution

Solutions to the problems | Forward to §3.7 | Back to §3.5 | Up | Table of Contents