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Abstract Algebra

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Kernel of a group homomorphism

ABSALG-JSXXEL

Let $G_1$ be a group with neutral element $e_1$ and let $G_2$ be a group with neutral element $e_2$.

Let $f:G_1\rightarrow G_2$ be a group homomorphism.

The kernel of $f$, denoted ${\rm Ker}(f)$, is the subset of $G_1$ given by

${\rm Ker}(f):=\{g_1\in G_1\mid f(g_1)=e_2\}$

Notation: $S_4$ is the group of permutations on $\{1,2,3,4\}$

Which of the following is FALSE?

A

There is a group homomorphism $f$ with domain $S_4$ and

${\rm Ker}(f)=\{e_1,(12),(34),(12)(34)\}$

B

There is a group homomorphism $f$ with domain $S_4$ such that:

$f(g_1)=f(g_1')$ if and only if $g_1H=g_1'H$

...where:

$H=\{e_1, (12)(34),(13)(24),(14)(23)\}$

C

There is no group homomorphism $f$ with domain $\{z\in{\mathbb C}\mid |z|=1\}$ and kernel

$\{z\in{\mathbb C}\mid |z|=1,\; z^3=1\;{\rm or}\; z^5=1\}$

D

Every subgroup of an abelian group $A$ is the kernel of a group homomorphism with domain $A$

E

For every $m\ge 1$ and every infinite cyclic group $C_\infty$, there is a group homomorphism $f$, with domain $C_\infty$, such that the index $[C_\infty:{\rm Ker}(f)]=m$