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9
votes
0answers
146 views
+100

$2$-group with two isomorphic normal subgroups of index $4$ with non-isomorphic quotients

Hypothesis: Let $P$ be a finite $2$-group with two isomorphic normal subgroups $M$ and $N$ such that $P/M\cong C_4$ (the cyclic group of order $4$) while $P/N\cong C_2^2$. By the lattice theorem, ...
5
votes
1answer
414 views

Groups in which normal subgroups consitiute a chain with respect to inclusion

Let G be a finite group. In general case, given two normal subgroups N and K of G, we need not to have N < K or K< N. The easiest example is the Klein 4-group V4 and its subgroups of order 2. So ...
22
votes
6answers
2k views

Existence of simultaneously normal finite index subgroups

It is well known that if $K$ is a finite index subgroup of a group $H$, then there is a finite index subgroup $N$ of $K$ which is normal in $H$. Indeed, one can observe that there are only finitely ...