Answer to Question #114001 in Abstract Algebra for Sourav Mondal

Question

Check whether any group of order 44 has a
proper normal subgroup or not.

Answer 1

Let $G$ be any group such that $|G|=44=2^2×11$ .

As $11\mid 44 \ \text{and} \ 11^2 \nmid 44$ therefore $G$ has a sylow 11-subgroup .

Let $n$ be the number of sylow 11-subgroup then

$n\equiv 1 \ mod (11)$ and $n\mid 4$ .

Therefore , $n=1,2 \ or \ 4$ .

But $n=1$ is the only solution of the congruence $n\equiv 1 \ mod (11)$

Hence $G$ has only one sylow 11-subgroup.

Again we known that only one sylow p-subgroup are Normal.

Therefore sylow 11-subgroup is Normal in $G$ .

Hence any group of order 44 has a proper normal subgroup.