Subgroups

• Mar 9th 2013, 12:25 PM
lovesmath
Subgroups
Let G be a group and H be a normal subgroup of G. If |H|=2, then show that H<=Z(G). I know I have to show that H is a subgroup of the center of G, but I don't know what to do.
• Mar 9th 2013, 01:24 PM
Lockdown
Re: Subgroups
The order of H is 2, so it has two elements. Because H is a normal subgroup, it is a subgroup in particular, hence it must contain the identity element.
Hence, H = {e,a} where a is some element of order 2 (there can only be one element of order 1, and the order of the non-trivial element must divide 2, hence it must be 2).

Now since H is a normal subgroup, by definition, $gag^{-1} \in H$ for all g in G. In particular, $gag^{-1}$ is either e or a.
Note that $a \mapsto gag^{-1}$ is an isomorphism (it is an automorphism, actually). This means that it must preserve the order of the element.
Hence, $gag^{-1}=a$, or else we would have a contradiction.

Thus, $gag^{-1} = a \Leftrightarrow ga=ag$ for all g in G.
Hence, a commutes with every element in G. Since e does this too, H is a subset of the center of G.
Clearly, since a has order 2, this is a subgroup of the center of G.
• Mar 9th 2013, 01:29 PM
ILikeSerena
Re: Subgroups
Quote:

Originally Posted by lovesmath
Let G be a group and H be a normal subgroup of G. If |H|=2, then show that H<=Z(G). I know I have to show that H is a subgroup of the center of G, but I don't know what to do.

Hi lovesmath! :)

If |H|=2, then we can write H={e,h}, which we know is a normal subgroup of G.

Is e an element of Z(G)?
Is h?
What are the requirements to call a set a subgroup? Are they satisfied in this case?
• Mar 10th 2013, 09:51 AM
lovesmath
Re: Subgroups
Both e and h are elements of Z(G), and every element in the center commutes with all other elements. So, e and h commute with each other since they are the only elements in H. Thus, H is a subgroup of Z(G).
• Mar 10th 2013, 11:54 AM
ILikeSerena
Re: Subgroups
You're mixing things up a bit.

Quote:

Originally Posted by lovesmath
Both e and h are elements of Z(G),

You don't know yet whether both e and h are elements of Z(G).
That's what we're trying to find out.
You're not supposed to simply state it.

Quote:

and every element in the center commutes with all other elements.
Yes that is the definition of Z(G).
Good!

Quote:

So, e and h commute with each other since they are the only elements in H.
Yes, e and h commute with each other.
Not because they are the only elements in H, but simply because they are elements of a normal subgroup.

Quote:

Thus, H is a subgroup of Z(G).
I'm afraid this does not follow from the previous.