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 DanielThrice November 6th, 2010 10:25 AM

Subgroups

Original Question: Let G be an abelian group of order 540. What is the largest possible number of subgroups of order 3 such a group G can have?

My attempt: The fundamental theorem of group theory says that any group G of order 540 can be expressed as

G1 X G2

where G1 is a group of order 27 and G2 is a group of order 20.

If I'm searching for every group of order 3, then I guess I'm actually searching for every element of order 3. (number of subgroups of order 3= number of elements of order 3 divides by 2). Since an element of order 3 can only be in G1, I need to be searching for elements of order 3 in a group of order 27. This will require classifying all groups of order 27, but I don't know how to classify these groups. Help?

 skipjack November 25th, 2010 03:28 PM

Quote:
 Originally Posted by DanielThrice I'm actually searching for every element of order 3.
No, you're trying to find the maximum number of elements of order three in "such a group". It's possible that each non-identity element of G1 has order 3, so that maximum is 26, and so the maximum number of subgroups of order 3 is 13.

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