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 Post subject: Finite groupPosted: Mon, 27 Feb 2012 20:00:59 UTC

Joined: Sat, 26 Feb 2011 09:13:43 UTC
Posts: 5
Find all finite groups which satisfy the following conditions:

1) is not divisible by ;
2) has exactly cyclic subgroups.

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 Post subject: Re: Finite groupPosted: Mon, 27 Feb 2012 22:11:18 UTC
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Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 12097
Location: Austin, TX
Top Cat wrote:
Find all finite groups which satisfy the following conditions:

1) is not divisible by ;
2) has exactly cyclic subgroups.

Think about Cauchy's theorem. Also, be sure you know if you mean proper subgroups or not.

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 Post subject: Re: Finite groupPosted: Tue, 28 Feb 2012 05:21:48 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6006
Location: 127.0.0.1, ::1 (avatar courtesy of UDN)
Top Cat wrote:
Find all finite groups which satisfy the following conditions:

1) is not divisible by ;
2) has exactly cyclic subgroups.

Hint: which cyclic group(s) has/have exactly 2 generators?

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 Post subject: Re: Finite groupPosted: Tue, 28 Feb 2012 17:29:48 UTC

Joined: Sat, 26 Feb 2011 09:13:43 UTC
Posts: 5
outermeasure wrote:
Top Cat wrote:
Find all finite groups which satisfy the following conditions:

1) is not divisible by ;
2) has exactly cyclic subgroups.

Hint: which cyclic group(s) has/have exactly 2 generators?

Could you detail a bit please ?
I think only cyclic groups of order 3 satisfy the given conditions, but I don't know how to show that there are no other ones ...

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 Post subject: Re: Finite groupPosted: Tue, 28 Feb 2012 17:40:35 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6006
Location: 127.0.0.1, ::1 (avatar courtesy of UDN)
Top Cat wrote:
outermeasure wrote:
Top Cat wrote:
Find all finite groups which satisfy the following conditions:

1) is not divisible by ;
2) has exactly cyclic subgroups.

Hint: which cyclic group(s) has/have exactly 2 generators?

Could you detail a bit please ?
I think only cyclic groups of order 3 satisfy the given conditions, but I don't know how to show that there are no other ones ...

No. There are 3 cyclic groups that have exactly 2 elements as possible generators.

Also you would need the case of unique generator...

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