0 Definitions

Regular and semiregular

Def. (regular and semiregular) Suppose $G$ is a group acting on $\Omega$. If $G_{\alpha }=\{1\}$ for any $\alpha \in \Omega$, we say $G$ is semi-regular. if the group action is also transitive, $G$ is regular.

Prop. If $X$ is a finite set, let $B=\mathrm{Sym}(X)$, a subgroup $N\le B$ is regular if any two of the following conditions hold:

• $N$ acts transitively on $X$
• $N$ acts without fixed points
• $|N|=|X|$

A semi-regular subgroup of order $|X|$ is therefore automatically regular, and any semi-regular subgroup can have at most $|X|$ elements.

We also observe that any subgroup of $B$ which is transitive must have at least $|X|$ elements.

It’s relatively easy to show that a semi-regular subgroup $K\le B$ is a subgroup of a regular subgroup.

Does $M$ being transitive imply that it contains a regular subgroup? The answer is no.

Faithful Group Action

Faithful

A faithful group action is one where distinct elements of the group induce distinct permutations of the set being acted upon. In other words:

• No group elements (except the identity element) act the same way.
• The associated permutation representation is injective.
• Different elements of the group act differently at every point.

Core

Def. For a subgroup $H<G$, core of $H$ in $G$ is the largest subgroup of $H$ which is normal in $G$, i.e.$$ Core_GH=\cap_{g\in G}H^g.