This problem is from Seth Warner's Modern Algebra; problem number 12.21 (so Google can find it.) It's actually in the free preview, find it http://books.google.com/books?id=jd... Algebra "12.21"&pg=PA90#v=onepage&q&f=false"(adsbygoogle = window.adsbygoogle || []).push({});

1. The problem is stated as:

If h is an endomorphism of a group G such that [tex]\kappa[/tex]_{a}[tex]\circ[/tex] h = h [tex]\circ[/tex] [tex]\kappa[/tex]_{a}for every a [tex]\in[/tex] G, then the set H = {x[tex]\in[/tex]G : h(x) = h(h(x))} is a normal subgroup of G, and G/H is an abelian group.

2. [tex]\kappa[/tex]_{a}is the inner automorphism defined by a, and is defined as

[tex]\kappa[/tex]_{a}(x) = a[tex]\Delta[/tex]x[tex]\Delta[/tex]a*

where [tex]\Delta[/tex] is the group's binary operator and a* is the inverse of a.

We are also given a theorem that states that G/H is abelian iff [tex]x\Delta y\Delta x*[/tex][tex]\Delta y* \in H \forall x, y \in G [/tex] (which I also had to prove, but I'll spare you.)

3. Now, I managed to prove that H is a subgroup and that H is normal. I am at a loss, however, as to how to prove that G/H is abelian. I understand that I'm supposed to show that x[tex]\Delta[/tex]y[tex]\Delta[/tex]x*[tex]\Delta[/tex]y* [tex]\in[/tex] H [tex]\forall[/tex] x,y , but I can only see a way to do that if either x or y is in H, for example:

if x is in H, then

and since y[tex]\Delta[/tex]H[tex]\Delta[/tex]y* = H [tex]\forall[/tex] y because H is normal,

x[tex]\Delta[/tex]H = Hwhich contains x[tex]\Delta[/tex]y[tex]\Delta[/tex]x*[tex]\Delta[/tex]y* because H contains x*.

x[tex]\Delta[/tex](y[tex]\Delta[/tex]H[tex]\Delta[/tex]y*) = H

but the challenge is to show that this is true [tex]\forall[/tex] x,y[tex]\in[/tex]G

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Showing that a factor group is abelian

**Physics Forums | Science Articles, Homework Help, Discussion**