Fusion systems for dihedral group:D8

This article gives specific information, namely, fusion systems, about a particular group, namely: dihedral group:D8.
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This article discusses the possible fusion systems for the dihedral group of order eight.

$G=⟨a,x\mid a^4=x^2=e,xa{x}^{-1}=a^{-1}⟩$.

There are, up to isomorphism, two possible fusion systems on $G$.

The inner fusion system: the fusion system obtained from inner automorphisms

This is the fusion system where all morphisms are obtained as the restriction of inner automorphisms of $G$. The isomorphisms are as follows.

Isomorphisms between subgroups of order one

The trivial map. (1)

Isomorphisms between subgroups of order two

The following isomorphisms are included:

• All four identity maps for the subgroups $⟨x⟩,⟨ax⟩,⟨a^{2}x⟩,⟨a^{3}x⟩$. (4)
• The unique isomorphism between $⟨x⟩$ and $⟨a^{2}x⟩$. (1)
• The unique isomorphism between $⟨ax⟩$ and $⟨a^{3}x⟩$. (1)

Isomorphisms between subgroups of order four

The following isomorphism are included:

• The identity maps on all three subgroups: $⟨a⟩,⟨a^2,x⟩,⟨a^2,ax⟩$. (3)
• The cube map on the subgroup $⟨a⟩$. (1)
• The isomorphism of $⟨a^2,x⟩$ that fixes $a^2$ and interchanges $x$ and $a^{2}x$. (1)
• The isomorphism of $⟨a^2,ax⟩$ that fixes $a^2$ and interchanges $ax$ and $a^{3}x$. (1)
• The map $⟨a^2,x⟩\to ⟨a^2,ax⟩$ sending $a^2$ to itself and $x$ to $ax$.
• The map $⟨a^2,x⟩\to ⟨a^2,ax⟩$ sending $a^2$ to itself and $x$ to $a^{3}x$. (1)

Isomorphisms between subgroups of order eight

All the inner automorphisms:

• The identity map. (1)
• The map sending $a$ to itself and $x$ to $a^{2}x$. (1)
• The map sending $a$ to $a^3$ and $x$ to itself. (1)
• The map sending $a$ to $a^3$ and $x$ to $a^{2}x$. (1)

The other fusion system

This is unique up to automorphisms of the group. There are in fact two versions of this viewed strictly, which are interchanged under the outer automorphism sending $x$ to $ax$ and fixing $a$.

This fusion system is realized, for instance, in the symmetric group of degree four.

Isomorphisms between subgroups of order one

The trivial map. (1)

Isomorphisms between subgroups of order two

• All possible isomorphisms among the subgroups $⟨x⟩,⟨a^2⟩,⟨a^{2}x⟩$. This includes the three identity maps and the isomorphisms between distinct subgroups. (6)
• Identity maps on the subgroups $⟨ax⟩,⟨a^{3}x⟩$. (2)
• The isomorphism $⟨ax⟩\to ⟨a^{3}x⟩$. (1)

Isomorphisms between subgroups of order four

• All automorphisms of $⟨a^2,x⟩$. (6)
• The identity map on $⟨a⟩$. (1)
• The automorphism of $⟨a⟩$ sending $a$ to $a^3$. (1)
• The identity map on $⟨a^2,ax⟩$. (1)
• The isomorphism of $⟨a^2,ax⟩$ sending $ax$ to $a^{3}x$. (1)

Isomorphisms between subgroups of order eight

These are just the inner automorphisms of the whole group:

• The identity map. (1)
• The map sending $a$ to itself and $x$ to $a^{2}x$. (1)
• The map sending $a$ to $a^3$ and $x$ to itself. (1)
• The map sending $a$ to $a^3$ and $x$ to $a^{2}x$. (1)