Normality-preserving endomorphism-invariant subgroup
From Groupprops
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Definition
A subgroup of a group
is termed a normality-preserving endomorphism-invariant subgroup if, for every normality-preserving endomorphism
of
,
is contained in
. A normality-preserving endomorphism is an endomorphism with the property that the image of any normal subgroup is normal.
Examples
Extreme examples
- The trivial subgroup is normality-preserving endomorphism-invariant in any group.
- Every group is normality-preserving endomorphism-invariant in itself.
Examples arising from stronger properties or subgroup-defining functions
- All fully invariant subgroups, including the derived subgroup (commutator subgroup), as well as members of the derived series and lower central series, are normality-preserving endomorphism-invariant.
- The Fitting subgroup and solvable radical are both normality-preserving endomorphism-invariant. In fact, they are something stronger: weakly normal-homomorph-containing subgroups.
Examples in small finite groups
Below are some examples of a proper nontrivial subgroup that satisfy the property normality-preserving endomorphism-invariant subgroup.
Group part | Subgroup part | Quotient part | |
---|---|---|---|
Cyclic maximal subgroup of dihedral group:D8 | Dihedral group:D8 | Cyclic group:Z4 | Cyclic group:Z2 |
Below are some examples of a proper nontrivial subgroup that does not satisfy the property normality-preserving endomorphism-invariant subgroup.
Group part | Subgroup part | Quotient part | |
---|---|---|---|
Diagonally embedded Z4 in direct product of Z8 and Z2 | Direct product of Z8 and Z2 | Cyclic group:Z4 | Cyclic group:Z4 |
This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]