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Subgroup order question

Posted: Fri Dec 17, 2010 9:07 am
by brain
If G is a group of order 12, then G must have a subgroup of all the following orders except

A 1 B 2 C 4 D 6 E 12

Is it evident that subgroup of order 4 always exists?

Re: Subgroup order question

Posted: Fri Dec 17, 2010 9:29 am
by johnnybegoog
Yeah by one of the Sylow theorems (http://planetmath.org/encyclopedia/SylowTheorems.html) G must have a 2-subgroup (ie here a subgroup of order 4), so the correct answer is D.

Re: Subgroup order question

Posted: Fri Dec 17, 2010 2:44 pm
by aaaaa
This is just their way of asking you to identify the fact that A_4 (which has 12 elements) has no subgroup of order 6, so if you were to interpret a potential "converse" to Lagrange's theorem in this way, it would be false.

Re: Subgroup order question

Posted: Sat Dec 18, 2010 6:53 am
by bobn
then G must have a subgroup

so, there might a subgroup of order 6, but its not true that there must be a subgroup of order 6.

Re: Subgroup order question

Posted: Mon Dec 20, 2010 9:23 am
by brain
OK, thanks everybody.