Math Is Fun Forum

matt00hew

Member

Registered: 2009-06-07

Posts: 2

Group of order 512

I'm interested in a problem that has turned out to be a group of order 512. Abelian and non cyclic. Is there a catalogue somewhere so I can find out which group it is? Or tests to do to narrow down the options?

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Group of order 512

If you want Abelian, that’s easy. Any non-cyclic Abelian group is isomorphic to one of the following:

The result you need is the fundamental theorem of finite Abelian groups.

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Group of order 512

Also

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Why did the vector cross the road?
It wanted to be normal.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Group of order 512

This is where

is the integers modulo n.  Jane, why the notation with C?  I typically see this in applications to physics and chemistry, but normally the group theoretic:

Or the number theoretic:

notations are used by mathematicians.  Just curious where you got it from.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

matt00hew

Member

Registered: 2009-06-07

Posts: 2

Re: Group of order 512

Thank you! That's incredibly helpful. Now I've seen your list it all makes sense.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Group of order 512

No, no.

is not

. It’s the cyclic group of order

. This notation is used by Humphreys.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Group of order 512

Jane,

is the cyclic group of order n.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."