Find all groups of order 121 up to isomorphism

Asked Dec 05 '13 at 07:59

Active Dec 05 '13 at 08:04

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Find all groups of order $121$ up to isomorphism.

if i'm right, all such groups are abelian groups, since this is $p^2$ for $p$ = $11$ and $11$ is prime.

there's only one, isn't there? isn't it just $\mathbb{Z}_{121} $ which is isomorphic to $\mathbb{Z}_{11} \times \mathbb{Z}_{11}$ ?

asked Dec 05 '13 at 07:59

furashu

2

No, there's more than one. Note $\Bbb Z/121 \Bbb Z$ is cyclic, while the highest order in $(\Bbb Z/11\Bbb Z)^2$ is 11 – Tim Ratigan Dec 05 '13 at 08:01
can you explain why? – furashu Dec 05 '13 at 08:02
Can you show that $\Bbb{Z}{121}$ and $\Bbb{Z}{11} \times \Bbb{Z}_{11}$ are not isomorphic? – Sammy Black Dec 05 '13 at 08:05
$Sammy wait, they're NOT isomorphic? Well no, I guess I couldn't... – furashu Dec 05 '13 at 08:08
@Ryan: Tim has given you a hint. – Dec 05 '13 at 08:10
how does one determine whether or not two groups are isomorphic, then? – furashu Dec 05 '13 at 08:13
Well, properties of isomorphism of two groups can help you to prove that they are NOT isomorphic. Some of these properties are : the group will have same order, they will have same number of elements of a fixed order; they will have same number of subgroups etc. – KON3 Dec 05 '13 at 08:18

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