Non-normal subgroups of M16
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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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.
|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 which is normal|
|subnormal subgroup||normal subgroup of normal subgroup of ... normal subgroup||Yes||follows from being 2-subnormal, also from being a subgroup of nilpotent group|