# Tagged: product of groups

## Problem 544

Let $G$ a finite group and let $H$ and $K$ be two distinct Sylow $p$-group, where $p$ is a prime number dividing the order $|G|$ of $G$.

Prove that the product $HK$ can never be a subgroup of the group $G$.

## Problem 493

Let $G$ be a finite group and let $A, B$ be subsets of $G$ satisfying
$|A|+|B| > |G|.$ Here $|X|$ denotes the cardinality (the number of elements) of the set $X$.
Then prove that $G=AB$, where
$AB=\{ab \mid a\in A, b\in B\}.$

## Problem 448

Let $G$ be a group. Let $H$ be a subgroup of $G$ and let $N$ be a normal subgroup of $G$.
The product of $H$ and $N$ is defined to be the subset
$H\cdot N=\{hn\in G\mid h \in H, n\in N\}.$ Prove that the product $H\cdot N$ is a subgroup of $G$.

## 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
$f^{-1}(f(H))=HN.$