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Let me denote the number of conjugacy classes by $k(G)$, since this seems to be more common. Let $G$ be a finite group, and let $p$ be a prime number that does not divide $|G|$. Suppose that $V$ is a …

As pointed out by Michael Giudici the answer is given by a result of Horoševskiĭ. Here is a proof following the paper by Horoševskiĭ. Lemma: Let $\phi$ be an automorphism of $G$ with $|\phi|$ divisib …

Let $\operatorname{gnu}(n)$ be the number of finite groups of order $n$. Question: Is it true that $\operatorname{gnu}(4n) \geq 2 \operatorname{gnu}(n)$ for all $n \geq 1$? Surely this must be true, …

The extraspecial group $P = 2_{+}^{1+2n}$ has generators $z$, $v_1$, $\ldots$, $v_n$, $w_1$, $\ldots$, $w_n$, with $Z(P) = \langle z \rangle$ cyclic of order $2$ and \begin{align*} v_i^2 = w_i^2 = 1 & …

(Edit: the previous version of this answer was not correct, but I leave this here as a remark) Let $\sigma: G \rightarrow G$ be the Frobenius map corresponding to the field automorphism $x \mapsto x^q …

Let $G$ be a group and let $K/k$ be a field extension. Suppose that $V$ is a $kG$-module, and let $V_K = K \otimes_k V$ be the $KG$-module given by changing the base field. Is it true that $H^n(G,V_K) …