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For which $n$ is there a unique perfect group of order $n$? Are there infinitely many such $n$? Some guesses for infinite sequences of such $n$: $|\mathrm{PSL}(2,p)|$, $|\mathrm{SL}(2,p)|$, $|A_m|$, …

There must be no other subgroup of $H$ isomorphic to $K$. Certainly this property implies your property, as $g^{-1}Kg$ is always isomorphic to $K$. Conversely if $K'\leq H$ is isomorphic to $K$ let $ …

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 …

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 …

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 …

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: …

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} \ …

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 $n \leq 2m$ …

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 $ …

In the case $k=0$ this is a conjecture of Dixon. See http://people.math.carleton.ca/~jdixon/Prgrpth.pdf, Section 1.2. I don't know of any reason it should become obviously false for $k>0$, so I guess …

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, …

I'm trying to understand the proof of Kantor's Singer cycle theorem, which asserts that if $G$ is a subgroup of $\operatorname{GL}(n,q)$ containing a Singer cycle then $\operatorname{GL}(n/s,q^s) \leq …

Ah, point I was overlooking is that in this case, $|\Delta|$ must be prime: this comes out of the application of Burnside-Schur. To spell out the rest of the proof (no doubt one way of many), we know …

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 …

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 …