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Introduction

Many groups can be defined by certain group presentation for example the cyclic group , the group $\mathbb{Z}/ m\mathbb{Z}\times \mathbb{Z}/ n\mathbb{Z}$, the dihedral group $\ldots$ etc (see wikipedia page presentation of a group).

Question ?

There is a group that is definied as follows, $$V_{8n}=⟨a, b|a^{ 2n} = b^{ 4} = e, ba = a^{ −1} b ^{ −1} , b^{ −1} a = a ^{ −1} b⟩.$$

Does anyone know the name of the above group ? (the $V_{8n}$ group).

Example: The group $D_n $ with presentation $⟨s, r|s^{ 2 } = r^{ n} = e, sr = r^{ −1} s⟩$ is called the dihedral group.

Thanks in advance !

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    I am afraid you group is pretty trivial since $b^{-1} a = a^{-1} b$ we have $b^{-1} a = e$ and as such $b = a$ – 2017-02-08
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    @the_architect: $b^{-1}a = a^{-1}b$ is equivalent to $(b^{-1}a)^2 = e$, not to $b^{-1}a = e$ – 2017-02-08
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    Where does this group come up? – 2017-02-08
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    @MichaelBurr In the following article (section 4): http://www.quasigroups.eu/contents/download/2016/24_03.pdf and in several other articles – 2017-02-08
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    I don't think these groups have a common name in general. Even for $n=2$, this is SmallGroup(16,3), which is somewhat famous for not having a really nice name. They still have some nice structural property (for example, $b^2$ is central, $\langle a_2\rangle$ is normal, etc...) – 2017-02-08
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    @verret Thanks ! – 2017-02-09
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    With GAP I always obtain the semidirect product $(C_{2n} \times C_2) \rtimes C_2$; If you can consider that a "name". – 2017-02-09
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    @verret: I've been trying to prove thatt $a$ and $b^2$ commute, how do you prove that? – 2017-02-09
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    $a^{-1}b^2a=(a^{-1}b)(ba)=b^{-1}aa^{-1}b^{-1}=b^{-2}=b^2$. – 2017-02-09
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    You can then easily show that $\langle a,b^2\rangle$ is an abelian normal subgroup of index $2$, and $ba$ is an element of order $2$ not contained in it, so you get a semi-direct product, as Marc said. – 2017-02-09
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    This is not the dicyclic group of order $4n$, but is pretty close to it! See [here](https://en.wikipedia.org/wiki/Dicyclic_group). – 2017-02-10

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