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.
- The Frattini subgroup of \(G\), \(\Phi(G)\), is the
intersection of all maximal subgroups of \(G\).
- Let \(Z(G)\) be the center of \(G\) and \(G'\), the commutator
subgroup of \(G\).
- \(G\) 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 \(m < n \).
- 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
\(Z(G/K)\).
- \(G\) is said to be nilpotent if it has a series all of whose factors are central factors of
\(G\).
- \(Z_n(G)\) 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 \(Z_n(G)=G\).
- \(G\) 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.
- \(G\) is an extra-special \(p\)-group if \(G\) is special and \(G'\) is cyclic of order \(p\).
- A group is divisible if each element is divisible by each positive integer.
- 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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