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Top Cat
 Post subject: Finite group
PostPosted: Mon, 27 Feb 2012 20:00:59 UTC 
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Find all finite groups G which satisfy the following conditions:

1) |G| is not divisible by 4;
2) G has exactly |G|-1 cyclic subgroups.
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Shadow
 Post subject: Re: Finite group
PostPosted: Mon, 27 Feb 2012 22:11:18 UTC 
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Top Cat wrote:
Find all finite groups G which satisfy the following conditions:

1) |G| is not divisible by 4;
2) G has exactly |G|-1 cyclic subgroups.


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

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outermeasure
 Post subject: Re: Finite group
PostPosted: Tue, 28 Feb 2012 05:21:48 UTC 
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Top Cat wrote:
Find all finite groups G which satisfy the following conditions:

1) |G| is not divisible by 4;
2) G has exactly |G|-1 cyclic subgroups.


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

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\begin{aligned} Spin(1)&=O(1)=\mathbb{Z}/2&\quad&\text{and}\\ Spin(2)&=U(1)=SO(2)&&\text{are obvious}\\ Spin(3)&=Sp(1)=SU(2)&&\text{by }q\mapsto(\mathop{\mathrm{Im}}\mathbb{H}\ni p\mapsto qp\bar{q})\\ Spin(4)&=Sp(1)\times Sp(1)&&\text{by }(q_1,q_2)\mapsto(\mathbb{H}\ni p\mapsto q_1p\bar{q_2})\\ Spin(5)&=Sp(2)&&\text{by }\mathbb{HP}^1\cong S^4_{round}\hookrightarrow\mathbb{R}^5\\ Spin(6)&=SU(4)&&\text{by the irrep }\Lambda_+\mathbb{C}^4 \end{aligned}
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Top Cat
 Post subject: Re: Finite group
PostPosted: Tue, 28 Feb 2012 17:29:48 UTC 
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Joined: Sat, 26 Feb 2011 09:13:43 UTC
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outermeasure wrote:
Top Cat wrote:
Find all finite groups G which satisfy the following conditions:

1) |G| is not divisible by 4;
2) G has exactly |G|-1 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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outermeasure
 Post subject: Re: Finite group
PostPosted: Tue, 28 Feb 2012 17:40:35 UTC 
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Top Cat wrote:
outermeasure wrote:
Top Cat wrote:
Find all finite groups G which satisfy the following conditions:

1) |G| is not divisible by 4;
2) G has exactly |G|-1 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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\begin{aligned} Spin(1)&=O(1)=\mathbb{Z}/2&\quad&\text{and}\\ Spin(2)&=U(1)=SO(2)&&\text{are obvious}\\ Spin(3)&=Sp(1)=SU(2)&&\text{by }q\mapsto(\mathop{\mathrm{Im}}\mathbb{H}\ni p\mapsto qp\bar{q})\\ Spin(4)&=Sp(1)\times Sp(1)&&\text{by }(q_1,q_2)\mapsto(\mathbb{H}\ni p\mapsto q_1p\bar{q_2})\\ Spin(5)&=Sp(2)&&\text{by }\mathbb{HP}^1\cong S^4_{round}\hookrightarrow\mathbb{R}^5\\ Spin(6)&=SU(4)&&\text{by the irrep }\Lambda_+\mathbb{C}^4 \end{aligned}
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