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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: …

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 $n$: $|\mathrm{PSL}(2,p)|$, $|\mathrm{SL}(2,p)|$, $|A_m|$, …

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 …

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 …

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} \ …

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 …

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 …

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 …

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 …

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 $ …

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 …

No, there is not such a word, by the following two facts. The two elements $$A=\left(\begin{array}{cc}1&2\\0&1\end{array}\right),\quad B=\left(\begin{array}{cc}1&0\\2&1\end{array}\right)$$ of $\text …

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 …