Forward to §3.7 | Back to §3.5 | Up | Table of Contents | About this document

**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 *n*th
**dihedral group**,
denoted by D_{n}.

We can describe the nth dihedral group as

D_{n}=
{ a^{k}, a^{k}b |
0 k < n },

**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
x_{1},x_{2},**...**,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.

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^{-1}b.
In the problems in this section, we will use the following description
of the dihedral group D_{n} of order 2n.

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

**22.**
In the dihedral group D_{n},
show that ba^{i} = a^{n-i} b, for all
0 i < n.
*Solution*

**23.**
In the dihedral group D_{n},
show that each element of the form a^{i} b has order 2.
*Solution*

**24.**
In S_{4}, find the subgroup H generated by (1,2,3) and (1,2).
*Solution*

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

**26.**
Show that each element in A_{4}
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 S_{3},
in S_{4}, and in A_{4}.
*Solution*

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