Excerpted from Beachy/Blair, Abstract Algebra, 2nd Ed. © 1996

## § 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 Dn.

We can describe the nth dihedral group as

Dn= { ak, akb | 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 Sn is a subgroup of Sn.

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

Let n be the polynomial in n variables x1,x2,...,xn defined to be the product of all factors (xi - xj) in which 1 i < j n. Any permutation in Sn acts on n by permuting the subscripts.

Lemma 3.6.6. Any transposition in Sn changes the sign of n.

Theorem 3.6.7. A permutation in Sn 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 Sn. 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 Dn. 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 Dn of order 2n.

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

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

23. In the dihedral group Dn, 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 Dn, find the centralizer of a.     Solution

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

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