Presentation of $A_5 \times \Bbb Z_2$.

Question

I want presentation of the group $A_5\times \Bbb Z_2$ which is a group of order $120.$

I know the presentation of $A_5$ but not of product.

I tried it in GAP . In GAP its Atlas name is $2\times A_5$ but I can't type $\times $ in computer. Please given me its presentation and how I write it in GAP. Thank you very much.

There's no such thing as the presentation of a group; there is infinitely many! Also, it's fairly standard to get a presentation of the direct product of two groups, given a presentation of each. Search engines are some of your friends. — Shaun, Dec 30 '18 at 13:45
Thanks ... Please give me presentation of this group only ..thanks — Ymylife, Dec 30 '18 at 13:50

score 4 · Accepted Answer · edited Jan 01 '19 at 23:25

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Hint: If $\langle X\mid R\rangle$ is a presentation of $A_5$, then $$\langle X\cup\{z\}\mid \{z^2\}\cup R\cup\{xz=zx\mid x\in X\}\rangle$$ for some symbol $z\notin X$, is a presentation of $A_5\times\Bbb Z_2$, since $\Bbb Z_2$ has $\langle z\mid z^2\rangle$ as a presentation; see this.

This chapter of the GAP Reference Manual explains how to implement direct products in GAP.

edited Jan 01 '19 at 23:25

Olexandr Konovalov

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answered Dec 30 '18 at 13:55

Shaun

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Thanks for for quick reply ... I am trying to understand this answer ... – Ymylife Dec 30 '18 at 14:07
You're welcome, @Yogesh. – Shaun Dec 30 '18 at 14:09
NB: I made a slight, technical edit, @Yogesh. – Shaun Dec 30 '18 at 14:12
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thanks sir you explain every thing very nicely ... my final question is in GAP this group name is $2\times A5$? Am I right... – Ymylife Dec 30 '18 at 14:21
Again, you're welcome, @Yogesh; as to your final question: I don't know. The library of small groups in GAP is vast. However, once you have implemented some finitely presented group object G in GAP, you can use the code StructureDescription(G); to give you what the group name is. Note, though, that it might not always output something. There's only so much that can be stored. – Shaun Dec 30 '18 at 14:27
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In GAP, IsomorphismFpGroup will determine a presentation for a given group, IsomorphismFpGroupByGenerators a presentation o a chosen generating set. – ahulpke Dec 30 '18 at 14:36
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In GAP, you cannot generate groups from a name, but e.g. from Permutation generators, or as products of existing groups, e.g. DirectProduct. – ahulpke Dec 30 '18 at 14:37
@ahulpke thanks I will try to generate ... – Ymylife Dec 30 '18 at 15:25

Presentation of $A_5 \times \Bbb Z_2$.

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