# Math Help - commutes

1. ## commutes

If a group is non-abelian, then the identity is the only thing that commutes with everything right?

2. Originally Posted by sfspitfire23
If a group is non-abelian, then the identity is the only thing that commutes with everything right?
Not necessarily. Plz read this
Center (group theory) - Wikipedia, the free encyclopedia

Group is abelian IFF Z(G)=G

3. Originally Posted by sfspitfire23
If a group is non-abelian, then the identity is the only thing that commutes with everything right?
For example, $G = S_4 \times C_2$. The $C_2$ will commute with everything.

Also, every group of order $p^n$ has a non-trivial centre, $p$ prime, for instance in $D_8$ the rotation by 180 degree commutes with everything and is non-trivial.

4. In a non-abelian group that is generated by <a>, will the only thing that commutes be $e$ or will what you guys said still hold?

5. Originally Posted by sfspitfire23
In a non-abelian group that is generated by <a>, will the only thing that commutes be $e$ or will what you guys said still hold?

Any cyclic group will always be abelian. This follows from associative property.

6. A group generated by a single element is cyclic and (therefore) abelian. So there is no such thing as the non-abelian group generated by <a> , whatever a may be.

7. Originally Posted by sfspitfire23
In a non-abelian group that is generated by <a>, will the only thing that commutes be $e$ or will what you guys said still hold?
Let $G=$. Then for $g,h \in G$ arbitrary, $g=a^n, h=a^m, m,n \in \mathbb{Z}$. Can you show that $gh=hg$?

EDIT: 3 replies in 2 minutes? That must be some kind of record!

8. Originally Posted by Swlabr
Let $G=$. Then for $g,h \in G$ arbitrary, $g=a^n, h=a^m, m,n \in \mathbb{Z}$. Can you show that $gh=hg$?
oops....too many helpers

9. Since the centre of a group is a normal subgroup, you can be sure that in a non-abelian simple group the identity is the only thing that commutes with everything. The smallest such group is the alternating group on 5 symbols.