Count the number of sub conjugacy classes comprising a conjugacy class contained in a normal subgroup

Let G be a group, H \leq G a normal subgroup, and K \subseteq G a conjugacy class such that K \subseteq H. Prove that K is a union of [G : HC_G(x)] distinct H-conjugacy classes, each of equal size. Deduce that a conjugacy class in S_n which consists of even permutations is either a single A_n conjugacy class or a union of two A_n-conjugacy classes of equal size.

Considering the induced action of H on K by conjugation, K is a union of H-conjugacy classes. Now let H \cdot a, H \cdot \subseteq K be distinct H-conjugacy classes. Because K is a G-conjugacy class, we have g \cdot a = b for some g \in G. (\cdot denotes conjugation.) Now g \cdot (H \cdot a) = gH \cdot a = Hg \cdot a = H \cdot (g \cdot a) = H \cdot b. Thus H \cdot a and H \cdot b are conjugate as subsets of G, and we have |H \cdot a| = |H \cdot b|. That is, all H-conjugacy classes in K have the same cardinality. If we fix x \in K, then, |K| = m \cdot |H \cdot x| for some integer m, where m is the number of distinct H-conjugacy classes in K.

Now note that C_H(x) = H \cap C_G(x) \leq H \leq H C_G(x) \leq G and that C_H(x) \leq C_G(x) \leq G. We then have the following.

[G : H \cap C_G(x)] = [G : H \cap C_G(x)]
[G : C_G(x)] \cdot [C_G(x) : H \cap C_G(x)] = [G : H] \cdot [H : C_H(x)]
|K| \cdot [C_G(x) : H \cap C_G(x)] = [G : HC_G(x)] \cdot [HC_G(x) : H] \cdot |H \cdot x|
|K| = [G : HC_G(x)] \cdot |H \cdot x|

Since by the Second Isomorphism Theorem, [HC_G(x) : H] = [C_G(x) : H \cap C_G(x)].

Thus m = [G : HC_G(x)] is the number of H-conjugacy classes contained in K, and all these classes have the same cardinality.

Consider now A_n \vartriangleleft S_n. If K \subseteq A_n is a conjugacy class and \sigma \in K, then by the above argument K is a union of [S_n : A_n C_{S_n}(\sigma)] distinct A_n-conjugacy classes of the same cardinality. Recall that A_n is a maximal subgroup of S_n. Since A_n \leq A_nC_{S_n}(\sigma) \leq S_n, we have that A_nC_{S_n}(\sigma) is either A_n or S_n. Thus the number of A_n-conjugacy classes in K is either 1 or 2, and these classes have the same size.

Post a comment or leave a trackback: Trackback URL.

Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: