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You can just compute the spectrum exactly using the formula mentioned by Benjamin Steinberg: $$\sum_g \operatorname{ord}(g) \frac{\chi(g)}{\chi(1)}.$$ There are two or four characters of degree $1$ ac …

As Nick Gill points out, one can certainly have $|B| > |A|$ if you do not assume coprimality. If you assume coprimality then yes $|B| < |A|$. If $A_p$ is the Sylow $p$-subgroup of $A$ then $\DeclareMa …

By the classification of finite abelian groups we have $A \cong C_1 \oplus \cdots \oplus C_k$ for some cyclic groups $C_i = \langle g_i \rangle$ of prime power order $r_i$. Let $\chi: A \times A \to …

Converting my comment into an answer: Let $G = G_1 \ge G_2 \ge \cdots$ be the lower central series. Then $G_k/G_{k+1}$ is spanned by the left-normed commutators $[x_1, \dotsc, x_k]$ with $x_1, \dotsc, …

Let $G_1$ and $G_2$ be nonisomorphic Sylow-isomorphic groups. For example let $G_1 = C_6$ and $G_2 = S_3$. Then for any finite group $H$, the groups $G_1 \times H$ and $G_2 \times H$ are nonisomorphic …

Your general question seems too general. Here is a partial answer to your specific question. Let $V = \mathbb{F}^{2m}$ and $G = \mathrm{Sp}_{2m}(\mathbb{F})$. Consider some $vg \in VG$. We know that $ …

Here is a solution in the special case in which each lower central factor $\gamma_i(G) / \gamma_{i+1}(G)$ is elementary abelian. First consider the case in which $G$ is elementary abelian, written add …

If $A$ and $B$ are elementary abelian $2$-subgroups of $\mathrm{PGL}_n(\mathbf C)$ of rank $r$ then they lift uniquely to elementary abelian $2$-subgroups of $\mathrm{GL}_n(\mathbf C)$ of rank $r+1$ ( …

You seem to be aware of the answer to your own question, since you give the reference to the paper of Guralnick and Magaard, which classifies groups of minimal degree $\leq n/2$. Therefore $n \leq 2m$ …

This is more of an extended comment than an answer. I will determine all the abelian groups failing to act faithfully on at most $n/3$ points. Suppose $G$ is abelian, say $G = C_{q_1} \times C_{q_2} \ …

For $w$ a word in a free group $F_d$, we can consider the word map $\bar w : G^d \to G$. Let $P_w(G)$ denote the proportion of $d$-tuples $x \in G^d$ such that $\bar w (x) = 1$. Your notation is relat …

The minimal projective degrees (minimal degree of an irreducible representation of a central extension) of the finite classical groups are (famously) given by Tiep and Zalesskii [1]. Is there a conven …

Suppose $G$ is a finite simple group of order $n$ with a nontrivial representation of degree $d$. Then $G$ is isomorphic to a subgroup of $U(d)$. By Collins's sharp version of Jordan's theorem (https: …

A transitive set in $\mathbf{R}^n$ is a finite set with a transitive group of symmetries. I want to understand how subsets of a transitive set constrain the group. Let me start with the example of a t …

Is there a convenient list of the maximal abelian subgroups of the projective semilinear group $\mathrm{P\Gamma L}_3(K) \cong \mathrm{PGL}_3(K) \rtimes \mathrm{Gal}(K)$ for $K$ a finite field? This is …