5.4. Group Actions and Group Structure
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5. ACTIONS OF GROUPS (b) Another way is to recall that Z2 Z2 can be described as a group with four elements e; a; b; c, with each nonidentity element of order 2 and the product of any two nonidentity elements equal to the third. Show that any permutation of fa; b; cg determines an automorphism and, conversely, any automorphism is given by a permutation of fa; b; cg.
5.3.8. Describe the automorphism group of Zn Zn. (The description need not be quite as explicit as that of Aut.Z2 Z2/.) Can you describe the automorphism group of .Zn/k ?
5.4. Group Actions and Group Structure
In this section, we consider some applications of the idea of group actions to the study of the structure of groups.
Consider the action of a group G on itself by conjugation. Recall that the stabilizer of an element is called its centralizer and the orbit of an element is called its conjugacy class. The set of elements z whose conjugacy class consists of z alone is precisely the center of the group. If G is finite, the decomposition of G into disjoint conjugacy classes gives the equation
X
jGj
jGj D jZ.G/j C ;
jCent.g/j
g
where Z.G/ denotes the center of G, Cent.g/ the centralizer of g, and the sum is over representatives of distinct conjugacy classes in G nZ.G/. This is called the class equation.
Example 5.4.1. Let’s compute the right side of the class equation for the group S4. We saw in Example 5.1.17 that S4 has only one element in its center, namely, the identity. Its nonsingleton conjugacy classes are of sizes 6, 3, 8, and 6. This gives 24 D 1 C 6 C 3 C 8 C 6.
Consider a group of order pn, where p is a prime number and n a positive integer. Every subgroup has order a power of p by Lagrange’s theorem, so for g 2 G nZ.G/, the size of the conjugacy class of g, namely, jGj
;
jCent.g/j
is a positive power of p. Since p divides jGj and jZ.G/j 1, it follows that p divides jZ.G/j. We have proved the following:
Proposition 5.4.2. If jGj is a power of a prime number, then the center of G contains nonidentity elements.
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5.4. GROUP ACTIONS AND GROUP STRUCTURE
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We discovered quite early that any group of order 4 is either cyclic or isomorphic to Z2 Z2. We can now generalize this result to groups of order p2 for any prime p.
Corollary 5.4.3. Any group of order p2, where p is a prime, is either cyclic or isomorphic to Zp Zp.
Proof. Suppose G, of order p2, is not cyclic. Then any nonidentity element must have order p. Using the proposition, choose a nonidentity element g 2 Z.G/. Since o.g/ D p, it is possible to choose h 2 G n hgi.
Then g and h are both of order p, and they commute.
I claim that hgi \ hhi D feg. In fact, hgi \ hhi is a subgroup of hgi, so if it is not equal to feg, then it has cardinality p; but then it is equal to hgi and to hhi. In particular, h 2 hgi, a contradiction.
It follows from this that hgihhi contains p2 distinct elements of G, hence G D hgihhi. Therefore, G is abelian.
Now hgi and hhi are two normal subgroups with hgi \ hhi D feg and hgihhi D G. Hence G Š hgi hhi Š Zp Zp.
n
Look now at Exercise 5.4.1, in which you are asked to show that a group of order p3 (p a prime) is either abelian or has center of size p.
Corollary 5.4.4. Let G be a group of order pn, n > 1. Then G has a normal subgroup feg N G. Furthermore, N can be chosen so that
¤
¤
every subgroup of N is normal in G.
Proof. If G is nonabelian, then by the proposition, Z.G/ has the desired properties. If G is abelian, every subgroup is normal. If g is a nonidentity element, then g has order ps for some s 1. If s