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A group $G$ is called self-replicating, if there exists a finite index subgroup $H$, such that $H\cong G\times\dots\times G$. Maybe the most famous example of a self-replicating group is a subgroup $K …

In general not. Consider the homomorphism $\mathrm{Aff}(p)\rightarrow\mathbb{F}_p^*$ mapping $ax+b$ to $a$. The probability that two random elements generate a subgroup for which the induced map is st …

Let $\Gamma$ be a free group over the alphabet $X$, $|X|\geq 2$, and put $S=X\cup X^{-1}$. Pick an increasing sequence of integers $n_i$, and put $\Lambda=\{g\in\Gamma|\exists i:\ell(g)=n_i\}$. Under …

In general the problem is very difficult. There has been quite some work on the diameter of the Cayleygraph of $S_n$, the best results being due to Helfgott-Seress for the general case, and Helfgott-S …

It is well known that a finite 4-times transitive permutation group is Matthieu, symmetric, or alternating. Another way of stating this is that the set $$ \Omega = \{(x_1, x_2, x_3, x_4), 1\leq x_i\le …

No. There are Tarski monsters in which all proper subgroups are conjugated. I don't have a reference for this, but I read that Mann said that Rips said so.

One method that often works is the following: A primitive group, which contains a $p$-cycle for a prime $p<n-2$ is $A_n$ or $S_n$. If a permutation $\pi$ contains a cycle of length $p$, and no other c …

If you exclude trivial examples, which lead to girth 3, then for sufficiently large $n$ the answer is 4. Assume for simplicity that $n$ is odd. Suppose there exists a prime number $p$ with $n/2<p\le …

A nilpotent group is the direct product of its Sylow subgroups, hence the nilpotency class of a nilpotent group equals the maximum of the nilpotency classes of its Sylow subgroups. A group of order $p …

Pick a prime $p$ in the range $\frac{n}{2}<p\leq n-k$. Then $S_{n-k}$ contains $(p-1)!\binom{n-k}{p}$ elements of order $p$, and elements of order $p$ in $S_n$ are $p$-cycles. There are at most $(p-1) …

P. Hall gave a formula for the number of generators of $G^n$ for any finite simple group $G$. One famous example is the fact that $A_5^{19}$ is 2-generated, but $A_5^{20}$ is not. The question of comp …

Lackenby ( http://people.maths.ox.ac.uk/lackenby/lg070105.ps ) proved that a finitely presented group, which has a pro-$p$ completion of positive rank gradient, is large, i.e. it contains a subgroup o …

let $G$ be a transitive permutation group acting on $\{1, \ldots, n\}$, and let $d(G)$ be the minimal number of generators of $G$. Is it true, that for $n\rightarrow\infty$ we have $\frac{d(G)\log|G|} …

There is no such ordering, which is compatible with the profinite topology in the following way: If $x<y$, then there are small neighbourhoods $U, V$ of $x, y$, such that $u<v$ for all $u\in U, v\in V …

One area where logic really helped group theory is the theory of zetafunctions of torsionfree nilpotent groups. Define $\zeta_G(s)=\sum_{U\leq G} (G:U)^{-s}$, where summation runs over all finite inde …