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I'm trying to work through a sketch proof attributed to Walter Feit on characterizing $S_5$.

Suppose $G$ is a finite group with exactly two conjugacy classes of involutions, with $u_1$ and $u_2$ being representatives. Suppose $C_1=C(u_1)\simeq \langle u_1\rangle\times S_3$ and $C_2=C(u_2)$ be a dihedral group of order $8$. The eventual result is that $G\simeq S_5$. Also, $C(u)$ denotes the centralizer of $u$ in $G$.

The sketch says "$C_2$ contains three classes of involutions. If $x$ is an involution in $C_2$, $x\neq u_2$ then $x$ is conjugate to $xu_2$. Why does $C_2$ contain three classes of involutions, and why is any involution $x\in C_2$ such that $x\neq u_2$ conjugate to $xu_2$?


My thoughts: From the answer here, I know that $C_2$ is a Sylow $2$-subgroup. Since $u_1$ is contained in a Sylow $2$-subgroup, and all the Sylow $2$-subgroups are conjugate to each other, I can replace $u_1$ by some conjugate, and without loss of generality assume that $u_1\in C_2$. Then $u_2\in C_1$ as well.

Then I know $C_2$ contains at least two classes for $u_1$ and $u_2$. They can't be conjugate in $C_2$ lest they be conjugate in $G$. Thus $C_2$ has at least $2$ classes of involutions, but why are there exactly $3$? My guess is $u_1u_2$ might be the third class, but I can't make progress with this guess. Thanks.

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Hints: You can see that $C_{2}$ contains one element of order $1$ $2$ elements of order $4$ and $5$ elements of order $2$. The two elements of order $4$ are mutual inverses and are conjugate. One of the elements of order $2$ is n teh centre of $C_{2}$ and forms a conjugacy class of size $1$. Show that the other $4$ involutions fall into two conjugacy classes, each of size $2$. –  Geoff Robinson Jun 5 '12 at 23:03
    
Thanks Geoff, I didn't really exploit the dihedral structure of the centralizer enough. –  Adelaide Dokras Jun 6 '12 at 6:58
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up vote 2 down vote accepted

The dihedral group of order $8$, $D_8 = \langle r,s \mid r^4 = s^2 = 1, sr = r^3s\rangle$, has exactly five elements of order $2$: $r^2$, which is central, and $s$, $rs$, $r^2s$, and $r^3s$, which are not central.

Since $u_2\in C(u_2)$ and is central (since it commutes with everything in its centralizer), $u_2$ is a conjugacy class by itself, and corresponds to the element $r^2$ in $D_8$.

Now notice that the conjugates of $s$ are just $s$ and $r^2s$: conjugating by a power of $r$ we get $r^isr^{-i} = r^{2i}s$, so we either get $r^2s$ (when $i$ is odd) or $s$ (when $i$ is even). And conjugating by an element of the form $r^is$ we get $(r^is)s(sr^{-i}) = r^isr^{-i} = r^{2i}s$ again.

Similarly, the conjugates of $rs$ are $rs$ and $r^3s$. Now, notice that $r^3s = (rs)r^2$, $rs = (r^3s)r^2$, $s=(r^2s)r^2$, and $r^2s = sr^2$. So, in each case, if $x\in D_8$ is an involution different from $r^2$, then the other conjugate of $x$ is $xr^2$.

So this happens simply because you have a group isomorphic to the dihedral group of order $88$, and nothing to do with the fact that it is a centralizer of an involution of a group with exactly two conjugacy classes of involutions.

Note also that you will not necessarily have $u_1u_2\in C_2$: that would require $c_1$ and $c_2$ to centralize each other, and this will depend on the representatives you choose (since, in $S_5$, the two classes of involutions are the transpositions $(i,j)$, and the products of two transpositions $(r,s)(t,u)$; these may commute, e.g., $(1,2)$ commutes with $(1,2)(3,4)$; but they need not commute, as for example $(1,2)$ does not commute with $(1,3)(4,5)$ ).

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