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Description: This note contains some definitions from group theory that are used in $\cal M22$, and forms [62 C], [62 D], [67 D], Form 180, and Form 308$(p)$

Content:

This note contains some definitions from group theory that are used in $\cal M22$, and forms [62 C], [62 D], [67 D], Form 180, and Form 308$(p)$

Definition: Let $G$ be a group and let $p$ be a prime.

1. The Frattini subgroup of $G$, $\Phi(G)$, is the intersection of all maximal subgroups of $G$.
2. Let $Z(G)$ be the center of $G$ and $G'$, the commutator subgroup of $G$.
3. $G$ is said to be a $p$- group if for all $x\in G$ there is an $n\in\omega$ such that $x^{p^n}=1$. $G$ is said to have exponent $p$ if for every $x\in G$, $x^p=1$.  An Abelian group is said to be elementary if there is a prime $p$ such that every non-identity element has order $p$. $G$ is said to be divisible  if for each $a\in G$ and for each integer $m\ne 0$ the equation $x^m=a$ has a solution in $G$ and $G$ is called $n$-divisible if the equation $x^m=a$ has a solution for all $m < n$.
4. A factor $H/K$ of a series of $G$ is said to be a central factor of $G$ if $K$ is normal in $G$ and $H/K$ is a subgroup of $Z(G/K)$.
5. $G$ is said to be nilpotent if it has a series all of whose factors are central factors of $G$.
6. $Z_n(G)$ is defined as follows: $Z_0(G)=1$ and for each integer $n>0$, $$Z_n(G)/Z_{n-1}(G)=Z(G/Z_{n-1}(G)).$$ The class of $G$ is the least integer $n$ such that $Z_n(G)=G$.
7. $G$ is a special $p$-group if $G$ is a $p$-group, $G'=Z(H)=\Phi(H)$, and $G$ is either elementary Abelian or nilpotent of class 2.
8. $G$ is an extra-special $p$-group if $G$ is special and $G'$ is cyclic of order $p$.
9. A group is divisible if each element is divisible by each positive integer.
10. A group is torsion free if the only element of finite order is the identity element.

Howard-Rubin number: 24

Type: Definitions

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