# Homomorphisms

• Apr 12th 2012, 02:43 PM
Aryth
Homomorphisms
Let $G_1$ and $G_2$ be groups with identities $e_1$ and $e_2$, respectively. Let $f: G_1 \to G_2$ be a homomorphism of groups. Let $K$ be the kernel of $f$. Let $x \in K$. Which of the following is/are necessarily true?

a) $x = e_1$
b) $x = e_2$
c) $f(x) = e_2$
d) $f(x) = e_1$
e) $f(x) = e_1e_2$

I've never taken abstract algebra before so I have never seen anything like this. Any help would be greatly appreciated.
• Apr 12th 2012, 02:47 PM
JSB1917
Re: Homomorphisms
The Kernel takes an element from the first group and maps it to the identity in the second group.
• Apr 13th 2012, 03:42 PM
Deveno
Re: Homomorphisms
by definition, $ker(f) = \{g \in G_1: f(g) = e_2\}$. so clearly (c) is always true.

(a) is sometimes true, but not always. for example, let $G_1 = \mathbb{Z}$ and $G_2 = \mathbb{Z}_5$. the map:

$f:\mathbb{Z} \to \mathbb{Z}_5$ given by $f(x) = x\ (\text{mod } 5)$ is a homomorphism, but $e_1 = 0$ and $5 \in ker(f)$,

but clearly $5 \neq 0$.

(b) is NEVER true. the kernel of f lies entirely in G1 (unless G1 is a subgroup of G2, in which case $e_1 = e_2$).

(d) is likewise NEVER true, f(x) is always in G2 (but see above).

(e) if G1, G2 are distinct groups, $e_1e_2$ isn't even defined.
• Apr 15th 2012, 06:33 AM
JSB1917
Re: Homomorphisms
Quote:

Originally Posted by Deveno
(a) is sometimes true, but not always. for example, let $G_1 = \mathbb{Z}$ and $G_2 = \mathbb{Z}_5$. the map:

$f:\mathbb{Z} \to \mathbb{Z}_5$ given by $f(x) = x\ (\text{mod } 5)$ is a homomorphism, but $e_1 = 0$ and $5 \in ker(f)$,

but clearly $5 \neq 0$.

Actually, I'm pretty sure the identity in the first group is always mapped to the identity in the second group. I believe that's a property of group homomorphisms.
• Apr 15th 2012, 02:09 PM
Deveno
Re: Homomorphisms
Quote:

Originally Posted by JSB1917
Actually, I'm pretty sure the identity in the first group is always mapped to the identity in the second group. I believe that's a property of group homomorphisms.

yes, but an element of the kernel K is not necessarily the identity. x COULD be the identity ((a) COULD be true), but it is not NECESSARILY true.
• Apr 16th 2012, 11:20 AM
JSB1917
Re: Homomorphisms
Quote:

Originally Posted by Deveno
yes, but an element of the kernel K is not necessarily the identity. x COULD be the identity ((a) COULD be true), but it is not NECESSARILY true.

Yeah, I misread it. I thought x was the only thing in the kernel. But a) would only be true if there was only one element, otherwise, as you noted, it could be many other things.