1. ## Homomorphisms

Let $\displaystyle G_1$ and $\displaystyle G_2$ be groups with identities $\displaystyle e_1$ and $\displaystyle e_2$, respectively. Let $\displaystyle f: G_1 \to G_2$ be a homomorphism of groups. Let $\displaystyle K$ be the kernel of $\displaystyle f$. Let $\displaystyle x \in K$. Which of the following is/are necessarily true?

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

I've never taken abstract algebra before so I have never seen anything like this. Any help would be greatly appreciated.

2. ## Re: Homomorphisms

The Kernel takes an element from the first group and maps it to the identity in the second group.

3. ## Re: Homomorphisms

by definition, $\displaystyle ker(f) = \{g \in G_1: f(g) = e_2\}$. so clearly (c) is always true.

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

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

but clearly $\displaystyle 5 \neq 0$.

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

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

(e) if G1, G2 are distinct groups, $\displaystyle e_1e_2$ isn't even defined.

4. ## Re: Homomorphisms

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

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

but clearly $\displaystyle 5 \neq 0$.
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.

5. ## Re: Homomorphisms

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.

6. ## Re: Homomorphisms

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.