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?
Subgroup order question

 Posts: 13
 Joined: Tue Nov 09, 2010 5:32 pm
Re: Subgroup order question
Yeah by one of the Sylow theorems (http://planetmath.org/encyclopedia/SylowTheorems.html) G must have a 2subgroup (ie here a subgroup of order 4), so the correct answer is D.
Re: Subgroup order question
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
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.
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
OK, thanks everybody.