Tagged: image of a group homomorphism
Problem 616
Suppose that $p$ is a prime number greater than $3$.
Consider the multiplicative group $G=(\Zmod{p})^*$ of order $p-1$.

(a) Prove that the set of squares $S=\{x^2\mid x\in G\}$ is a subgroup of the multiplicative group $G$.

(b) Determine the index $[G : S]$.

(c) Assume that $-1\notin S$. Then prove that for each $a\in G$ we have either $a\in S$ or $-a\in S$.

Read solution

Click here if solved 126 Add to solve later
Group Theory

03/20/2017

by
Yu
· Published 03/20/2017
· Last modified 07/07/2017

Problem 346
Let $G$ be a finite group of order $21$ and let $K$ be a finite group of order $49$.
Suppose that $G$ does not have a normal subgroup of order $3$.
Then determine all group homomorphisms from $G$ to $K$.

Read solution

Click here if solved 65 Add to solve later
Group Theory

12/03/2016

by
Yu
· Published 12/03/2016
· Last modified 08/11/2017

Problem 208
Let $G, G’$ be groups and let $f:G \to G’$ be a group homomorphism.
Put $N=\ker(f)$. Then show that we have
\[f^{-1}(f(H))=HN.\]

Read solution

Click here if solved 20 Add to solve later