Math Help - Algebra, Problems For Fun! (4)

1. Algebra, Problems For Fun! (4)

Ok, for this one you only need some very basic knowledge of group theory:

Let $G$ be a finite non-abelian group and $|G|=n.$ Let $k$ be the number of conjugacy classes of $G$. Prove that $k \leq \frac{5n}{8}$ with equality if and only if $|Z(G)|=\frac{n}{4}.$

2. Originally Posted by NonCommAlg
Ok, for this one you only need some very basic knowledge of group theory:

Let $G$ be a finite non-abelian group and $|G|=n.$ Let $k$ be the number of conjugacy classes of $G$. Prove that $k \leq \frac{5n}{8}$ with equality if and only if $|Z(G)|=\frac{n}{4}.$
Is $Z(G)$ the center of G? (I should know this, but it has been awhile.)

3. Originally Posted by chabmgph

s $Z(G)$ the center of G?
yes.

4. Originally Posted by NonCommAlg
Ok, for this one you only need some very basic knowledge of group theory:

Let $G$ be a finite non-abelian group and $|G|=n.$ Let $k$ be the number of conjugacy classes of $G$. Prove that $k \leq \frac{5n}{8}$ with equality if and only if $|Z(G)|=\frac{n}{4}.$
I will use the fact that the smallest non-abelian group has order 6, which is not hard to show.

Let $C(g_1),C(g_2),...,C(g_k)$ be the conjugacy classes of $G$. Assume WLOG that the first $m$ are non-central.

Suppose $|Z(G)|\not\le{\frac{n}{4}}.$ Then $|G/Z(G)|\lneq{4}$ Thus $|G/Z(G)|= 3,2, or 1.$ We will reach a contradiction for each case.

The case $|G/Z(G)|=1$ is impossible since this would imply $G$ were abelian. So assume $|G/Z(G)|=2$.

Let $g_i\in{G}$ with $i\le{m}$. Let $g\in{G\setminus{Z(G)}}$ with $g\neq{g_i}$. Consider $gg_{i}g^{-1}$

Since $g$ and $g_i$ are both noncentral and $G/Z(G)$ has order two we know that $gg_i\in{Z(G)}$

Thus $gg_ig^{-1} = g^{-1}gg_i=g_i$
Hence, we have shown that $gg_ig^{-1} = g_i$ for all $g\in{G}$ But this implies $g_i\in{Z(G)}$ which is a contradiction.

I don't have time to complete the last case. But it is fairly similar. Again use the fact that we know the group structure of $G/Z(G)$ and find the similar contradiction. Then use the class equation to get the inequality for $k$. I have not proven the case of equality but I'll leave that for someone else.

5. Originally Posted by bulls6x
I will use the fact that the smallest non-abelian group has order 6, which is not hard to show.

Let $C(g_1),C(g_2),...,C(g_k)$ be the conjugacy classes of $G$. Assume WLOG that the first $m$ are non-central.

Suppose $|Z(G)|\not\le{\frac{n}{4}}.$ Then $|G/Z(G)|\lneq{4}$ Thus $|G/Z(G)|= 3,2, or 1.$ We will reach a contradiction for each case.

The case $|G/Z(G)|=1$ is impossible since this would imply $G$ were abelian. So assume $|G/Z(G)|=2$.

Let $g_i\in{G}$ with $i\le{m}$. Let $g\in{G\setminus{Z(G)}}$ with $g\neq{g_i}$. Consider $gg_{i}g^{-1}$

Since $g$ and $g_i$ are both noncentral and $G/Z(G)$ has order two we know that $gg_i\in{Z(G)}$

Thus $gg_ig^{-1} = g^{-1}gg_i=g_i$
Hence, we have shown that $gg_ig^{-1} = g_i$ for all $g\in{G}$ But this implies $g_i\in{Z(G)}$ which is a contradiction.

I don't have time to complete the last case. But it is fairly similar. Again use the fact that we know the group structure of $G/Z(G)$ and find the similar contradiction. Then use the class equation to get the inequality for $k$. I have not proven the case of equality but I'll leave that for someone else.
you almost got the idea but you need to put everything together. a quick way to show that $|G/Z(G)| \geq 4$ is to recall that if G is non-abelian, then $G/Z(G)$ is never cyclic. now you need to

combine this result with the class equation to prove the first part. the proof of the second part of the problem comes from the proof of the first part.