5
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I constructed the non-abelian groups of order $16$ and listed the structure descriptions. The result was :

16
(C4 x C2) : C2
C4 : C4
C8 : C2
D16
QD16
Q16
C2 x D8
C2 x Q8
(C4 x C2) : C2

The group (C$4$ x C$2$) : C$2$ appears twice. Obviously, two non-isomorphic groups with this structure exist.

What are these groups and how do they differ ?

A similar result appears for order $20$ :

20
C5 : C4
C5 : C4
D20

What are the non-isomorphic groups with structure $C5:C4$ ?

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  • $\begingroup$ See 7.12: Can non-isomorphic groups have equal structure descriptions? from the GAP F.A.Q. $\endgroup$
    – Olexandr Konovalov
    Commented Jan 19, 2016 at 20:21
  • $\begingroup$ You can retrieve an operation table for those groups and hand analyze them if you really want to get to know those groups. The command for the operation table is MultiplicationTable(G); for given G. I've actually done that for all groups of order up to 18 (project I've worked on periodically with the aim of better understand of finite groups). $\endgroup$
    – Justin Benfield
    Commented Mar 13, 2016 at 20:51

1 Answer 1

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StructureDescription will -- despite what was claimed in older implementations -- not identify groups up to isomorphism, but just indicate a decomposition. For example $C_5$ has an automorphism group of order 4. So there are two semidirect (even if the spell-checker wants the word to be to be semidried) products $C_5:C_4$, namely one where $C_4$ acts as the automorphism of order 4, and one where it acts as the square of this automorphisms (that is the element of order 2 acts trivially).

Similar things happen in the other cases, E.g. if $C_4\times C_2=\langle a,b\rangle$ two different automorphisms of order 2 are $a\mapsto a^{-1}$ or $a\mapsto ab$ (both times fixing $b$), thus leading to non isomorphic semidirect products.

What this means is that one can use StructureDescription as an aid towards understanding a groups structure, but it is useless for determining isomorphism.

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    $\begingroup$ It is not really useless: you just used it to determine isomorphism classes with some extra information. $\endgroup$
    – Pedro ♦
    Commented Jan 18, 2016 at 23:49

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