Answer to Question #105104 in Abstract Algebra for Sourav Mondal

Question

Find the number of normal subgroups of order 25, and of order 50, of a group
of order 75.

Answer 1

By Lagrange's theorem, If $G$ is a finite group and $H$ is a subgroup of $G$ then order of $H$ divides the order of $G$ .

Hence, There is no normal subgroup of order 50 .

Now ,let $|G|=75$ and $|H|=25.$

Since ,$5^2\mid 75 \space and \space 5^3\nmid 75$ ,so $H$ is a sylow 5-subgroup.

Again, The number 'n' of sylow 5-subgroup of $G$ is 1 modulo 5 and divides 3.

Hence ,there is a unique sylow 5- subgroup.

Again we known that unique sylow 5-subgroup are normal.

Hence, H is the only normal subgroup of order 25 in $G.$