## Subgroup order question

Forum for the GRE subject test in mathematics.
brain
Posts: 28
Joined: Tue Aug 25, 2009 12:16 pm

### Subgroup order question

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?

johnnybegoog
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 2-subgroup (ie here a subgroup of order 4), so the correct answer is D.

aaaaa
Posts: 5
Joined: Sat Dec 11, 2010 6:20 pm

### 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.

bobn
Posts: 61
Joined: Thu Nov 12, 2009 2:59 am

### 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.

brain
Posts: 28
Joined: Tue Aug 25, 2009 12:16 pm

### Re: Subgroup order question

OK, thanks everybody.