Endomorphism structure of M16

This article gives specific information, namely, endomorphism structure, about a particular group, namely: M16.
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This article is about the structure of endomorphisms of M16 (GAP ID: (16,6)) which we will take as having the following presentation:

$⟨ a, x ∣ a^{8} = x^{2} = e, x a x = a^{5} ⟩$

where $e$ denotes the identity element.

Summary of information

Construct	Value	Order	Second part of GAP ID (if group)	Comment
endomorphism monoid	?	?	--
automorphism group	direct product of D8 and Z2	16	11
inner automorphism group	Klein four-group	4	2
outer automorphism group	Klein four-group	4	2
group of class-preserving automorphisms	Klein four-group	4	2	same as inner automorphism group
group of IA-automorphisms	Klein four-group	4	2	same as inner automorphism group
quotient of class-preserving automorphism group by inner automorphism group	trivial group	1	1
quotient of IA-automorphism group by inner automorphism group	trivial group	1	1
group of center-fixing automorphisms	dihedral group:D8	8	3
extended automorphism group	direct product of D8 and V4	32	46
quasiautomorphism group	?	?	?
1-automorphism group
holomorph
inner holomorph

Description of automorphism group

Inner automorphisms

For the actual description of automorphisms, we use the left action convention, so conjugation by $g$ is the map $h \mapsto g h g^{- 1}$ . With the right action convention, what we call conjugation by $g$ becomes conjugation by $g^{- 1}$ . However, the data in the table below is the same for both left and right action conventions because, since every square is in the center, every element and its inverse are in the same coset of the center.

For every inner automorphisms, there are four candidate elements that give rise to that inner automorphism via the action by conjugation. These elements together form a single coset of the center, which is the cyclic subgroup of order four generated by $a^{2}$ .

The columns below have been arranged so that elements of a conjugacy class are in adjacent columns to each other. From this, you can notice that the inner automorphisms permute elements within each conjugacy class.

Candidate elements by which conjugation describes the inner automorphism	Image of $e$	Image of $a^{4}$	Image of $a^{2}$	Image of $a^{6}$	Image of $a$	Image of $a^{5}$	Image of $a^{3}$	Image of $a^{7}$	Image of $a x$	Image of $a^{5} x$	Image of $a^{3} x$	Image of $a^{7} x$	Image of $x$	Image of $a^{4} x$	Image of $a^{2} x$	Image of $a^{6} x$
${e, a^{2}, a^{4}, a^{6}}$	$e$	$a^{4}$	$a^{2}$	$a^{6}$	$a$	$a^{5}$	$a^{3}$	$a^{7}$	$a x$	$a^{5} x$	$a^{3} x$	$a^{7} x$	$x$	$a^{4} x$	$a^{2} x$	$a^{6} x$
${a, a^{3}, a^{5}, a^{7}}$	$e$	$a^{4}$	$a^{2}$	$a^{6}$	$a$	$a^{5}$	$a^{3}$	$a^{7}$	$a^{5} x$	$a x$	$a^{7} x$	$a^{3} x$	$a^{4} x$	$x$	$a^{6} x$	$a^{2} x$
${a x, a^{3} x, a^{5} x, a^{7} x}$	$e$	$a^{4}$	$a^{2}$	$a^{6}$	$a^{5}$	$a$	$a^{7}$	$a^{3}$	$a x$	$a^{5} x$	$a^{3} x$	$a^{7} x$	$a^{4} x$	$x$	$a^{6} x$	$a^{2} x$
${x, a^{2} x, a^{4} x, a^{6} x}$	$e$	$a^{4}$	$a^{2}$	$a^{6}$	$a^{5}$	$a$	$a^{7}$	$a^{3}$	$a^{5} x$	$a x$	$a^{7} x$	$a^{3} x$	$x$	$a^{4} x$	$a^{2} x$	$a^{6} x$

The inner automorphism group is itself isomorphic to a Klein four-group. The multiplication table, viewed as cosets of the center, is as follows:

Element/element	${e, a^{2}, a^{4}, a^{6}}$	${a, a^{3}, a^{5}, a^{7}}$	${a x, a^{3} x, a^{5} x, a^{7} x}$	${x, a^{2} x, a^{4} x, a^{6} x}$
${e, a^{2}, a^{4}, a^{6}}$	${e, a^{2}, a^{4}, a^{6}}$	${a, a^{3}, a^{5}, a^{7}}$	${a x, a^{3} x, a^{5} x, a^{7} x}$	${x, a^{2} x, a^{4} x, a^{6} x}$
${a, a^{3}, a^{5}, a^{7}}$	${a, a^{3}, a^{5}, a^{7}}$	${e, a^{2}, a^{4}, a^{6}}$	${x, a^{2} x, a^{4} x, a^{6} x}$	${a x, a^{3} x, a^{5} x, a^{7} x}$
${a x, a^{3} x, a^{5} x, a^{7} x}$	${a x, a^{3} x, a^{5} x, a^{7} x}$	${x, a^{2} x, a^{4} x, a^{6} x}$	${e, a^{2}, a^{4}, a^{6}}$	${a, a^{3}, a^{5}, a^{7}}$
${x, a^{2} x, a^{4} x, a^{6} x}$	${x, a^{2} x, a^{4} x, a^{6} x}$	${a x, a^{3} x, a^{5} x, a^{7} x}$	${a, a^{3}, a^{5}, a^{7}}$	${e, a^{2}, a^{4}, a^{6}}$

Outer automorphisms and outer automorphism classes

The outer automorphism group is a Klein four-group, i.e., there are four cosets of the inner automorphism group in the automorphism group. Excluding the identity coset (which comprises the inner automorphisms) we thus get three outer automorphism classes. Each of these has size four (the size of the inner automorphism group) so there is a total of 12 outer automorphisms.

The automorphism group as a whole is isomorphic to direct product of D8 and Z2.

The outer automorphism group has a natural action as the normal V4 in S4 with its non-identity elements acting as double transpositions on the set of conjugacy classes ${{a, a^{5}}, {a^{3}, a^{7}}, {a x, a^{5} x}, {a^{3} x, a^{7} x}}$ . More explicitly:

One outer automorphism class interchanges the conjugacy class ${a, a^{5}}$ and ${a^{3}, a^{7}}$ and simultaneously also interchanges the conjugacy classes ${a x, a^{5} x}$ and ${a^{4} x, a^{7} x}$ .
One outer automorphism class interchanges the conjugacy classes ${a, a^{5}}$ and ${a x, a^{5} x}$ and simultaneously also interchanges the conjugacy classes ${a^{3}, a^{7}}$ and ${a^{3} x, a^{7} x}$ .
One outer automorphism class interchanges the conjugacy classes ${a, a^{5}}$ and ${a^{3} x, a^{7} x}$ and simultaneously also interchanges the conjugacy classes ${a^{3}, a^{7}}$ and ${a x, a^{5} x}$ .

Here now is a detailed description of all the 12 outer automorphisms in terms of their action on every element:

Automorphism class no. in above list	Order of automorphism	Quick description of automorphism	Image of $e$	Image of $a^{2}$	Image of $a^{4}$	Image of $a^{6}$	Image of $a$	Image of $a^{5}$	Image of $a^{3}$	Image of $a^{7}$	Image of $a x$	Image of $a^{5} x$	Image of $a^{3} x$	Image of $a^{7} x$	Image of $x$	Image of $a^{4} x$	Image of $a^{2} x$	Image of $a^{6} x$
1	2	$a \mapsto a^{3}, x \mapsto x$	$e$	$a^{4}$	$a^{6}$	$a^{2}$	$a^{3}$	$a^{7}$	$a$	$a^{5}$	$a^{3} x$	$a^{7} x$	$a x$	$a^{5} x$	$x$	$a^{4} x$	$a^{2} x$	$a^{6} x$
1	2	$a \mapsto a^{7}, x \mapsto x$	$e$	$a^{4}$	$a^{6}$	$a^{2}$	$a^{7}$	$a^{3}$	$a^{5}$	$a$	$a^{7} x$	$a^{3} x$	$a^{5} x$	$a x$	$x$	$a^{4} x$	$a^{2} x$	$a^{6} x$
1	2	$a \mapsto a^{3}, x \mapsto a^{4} x$	$e$	$a^{4}$	$a^{6}$	$a^{2}$	$a^{3}$	$a^{7}$	$a$	$a^{5}$	$a^{7} x$	$a^{3} x$	$a^{5} x$	$a x$	$a^{4} x$	$x$	$a^{6} x$	$a^{2} x$
1	2	$a \mapsto a^{7}, x \mapsto a^{4} x$	$e$	$a^{4}$	$a^{6}$	$a^{2}$	$a^{7}$	$a^{3}$	$a^{5}$	$a$	$a^{3} x$	$a^{7} x$	$a x$	$a^{5} x$	$a^{4} x$	$x$	$a^{6} x$	$a^{2} x$
2	2	$a \mapsto a x, x \mapsto x$	$e$	$a^{4}$	$a^{6}$	$a^{2}$	$a x$	$a^{5} x$	$a^{7} x$	$a^{3} x$	$a$	$a^{5}$	$a^{7}$	$a^{3}$	$x$	$a^{4} x$	$a^{2} x$	$a^{6} x$
2	2	$a \mapsto a^{5} x, x \mapsto x$	$e$	$a^{4}$	$a^{6}$	$a^{2}$	$a^{5} x$	$a x$	$a^{3} x$	$a^{7} x$	$a^{5}$	$a$	$a^{3}$	$a^{7}$	$x$	$a^{4} x$	$a^{2} x$	$a^{6} x$
2	4	$a \mapsto a x, x \mapsto a^{4} x$	$e$	$a^{4}$	$a^{6}$	$a^{2}$	$a x$	$a^{5} x$	$a^{7} x$	$a^{3} x$	$a^{5}$	$a$	$a^{3}$	$a^{7}$	$a^{4} x$	$x$	$a^{6} x$	$a^{2} x$
2	4	$a \mapsto a^{5} x, x, \mapsto a^{4} x$	$e$	$a^{4}$	$a^{2}$	$a^{6}$	$a^{5} x$	$a x$	$a^{3} x$	$a^{7} x$	$a$	$a^{5}$	$a^{7}$	$a^{3}$	$x$	$a^{4} x$	$a^{2} x$	$a^{6} x$
3	4	$a \mapsto a^{3} x, x \mapsto x$	$e$	$a^{4}$	$a^{2}$	$a^{6}$	$a^{3} x$	$a^{7} x$	$a^{5} x$	$a x$	$a^{3}$	$a^{7}$	$a^{5}$	$a$	$x$	$a^{4} x$	$a^{2} x$	$a^{6} x$
3	4	$a \mapsto a^{7} x, x \mapsto x$	$e$	$a^{4}$	$a^{2}$	$a^{6}$	$a^{7} x$	$a^{3} x$	$a x$	$a^{5} x$	$a^{7}$	$a^{3}$	$a$	$a^{5}$	$x$	$a^{4} x$	$a^{2} x$	$a^{6} x$
3	2	$a \mapsto a^{3} x, x \mapsto a^{4} x$	$e$	$a^{4}$	$a^{2}$	$a^{6}$	$a^{3} x$	$a^{7} x$	$a^{5} x$	$a x$	$a^{7}$	$a^{3}$	$a$	$a^{5}$	$a^{4} x$	$x$	$a^{6} x$	$a^{2} x$
3	2	$a \mapsto a^{7} x, x \mapsto a^{4} x$	$e$	$a^{4}$	$a^{2}$	$a^{6}$	$a^{7} x$	$a^{3} x$	$a x$	$a^{5} x$	$a^{3}$	$a^{7}$	$a^{5}$	$a$	$x$	$a^{4} x$	$a^{2} x$	$a^{6} x$