# 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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## Definition

We consider the group:

$G = M_{16} = \langle a,x \mid a^8 = x^2 = e, xax = a^5 \rangle$

with $e$ denoting the identity element.

This is a group of order 16, with elements:

$\! \{ e, a, a^2, a^3, a^4, a^5, a^6, a^7, x, ax, a^2x, a^3x, a^4x, a^5x, a^6x, a^7x \}$

We are interested in the following two conjugate subgroups:

$\! H_1 = \{ e, x \}, H_2 = \{e, a^4x \}$

The two subgroups are conjugate by any element not centralizing either of them. Specifically, we can choose any of the elements $a,a^3,a^5,a^7,ax,a^3x,a^5x,a^7x$ 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 $a$
2-subnormal subgroup normal subgroup of normal subgroup Yes normal inside $\langle a^4,x \rangle$ 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