Illustrate with proof of a finite group M with two normal su

Illustrate with proof of a finite group M with two normal subgroups N and K M/N M/K, N K.

Solution

Let M be the Dihedral gr²oup of order 8 generated by a (rotation ) of order 4 and x (reflection) of order 2 with the following presentation:

<x,a| a⁴ =x² =e, xax^-1 =a^-1 }

Note: a² commutes with every element of M.(a² x= x xa² x= xa^-2 =xa² )

Consider the subgroups

K={ e,x,a² ,a²x} (K is a subgroup because of the observation above).

Every ((non trivial) element of K is of order 2.

N= {e, a, a² ,a³ }

N is a cyclic group of order 4 (generated by a)

So K and N cant be isomorphic.

But M/K and M/N , both being groups of order 2 are necessarily isomorphic.