Non-normal subgroups of M16
From Groupprops
This article is about a particular subgroup in a group, up to equivalence of subgroups (i.e., an isomorphism of groups that induces the corresponding isomorphism of subgroups). The subgroup is (up to isomorphism) cyclic group:Z2 and the group is (up to isomorphism) M16 (see subgroup structure of M16).
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Definition
We consider the group:
with denoting the identity element.
This is a group of order 16, with elements:
We are interested in the following two conjugate subgroups:
The two subgroups are conjugate by any element not centralizing either of them. Specifically, we can choose any of the elements to conjugate either subgroup into the other.
Subgroup properties
Normality
Property | Meaning | Satisfied? | Explanation | Comment |
---|---|---|---|---|
normal subgroup | invariant under inner automorphisms | No | The two subgroups are conjugate to each other via ![]() |
|
2-subnormal subgroup | normal subgroup of normal subgroup | Yes | normal inside ![]() |
|
subnormal subgroup | normal subgroup of normal subgroup of ... normal subgroup | Yes | follows from being 2-subnormal, also from being a subgroup of nilpotent group |