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A central product of two groups $G$ and $H$ is determined as follows. The groups $G$ and $H$ have respective central subgroups $A$ and $B$ which are isomorphic, let $\delta:A\rightarrow B$ be such a …

First, let me include the same disclaimer that goes in the first line of any article I write: all groups considered herein are finite. Academically, I work with connecting the arithmetic structure of …

I certainly appreciate Max's answer (especially the geometric interpretation) and have accepted it as such. However, as is so usual, writing up the question caused me to focus on it with greater clar …

The background: We recall/define the following: $\Omega_n=\{1,\dots,n\}$. $M_n$ is the Mathieu group of degree $n$. We follow the Wikipedia article "Mathieu group" and define these groups for each …

For a straightforward example that the subgroup doing the acting in a semidirect product cannot automatically be cancelled (and as semidirect products are knit products, also addresses that case), the …

In the Wikipedia article Zappa–Szép product , the knit product (a.k.a. Zappa–Szép product, Zappa–Rédei-Szép product, general product, exact factorization) is defined, and its basic properties are laid …

By knit product (alias: Zappa-Szép product), I mean a product $AB$ of subgroups for which $A\cap B=1$. In particular, note that neither subgroup is required to be normal, thus making this a generaliz …