A group is an algebraic structure consisting of a set of elements together with an operation that satisfies four conditions: closure, associativity, identity and invertibility. Group theory studies groups.

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How to count the closed left-hand turn paths of planar bicubic graphs?

When you draw a planar cubic bipartite graph $\Gamma$ and 3-color its edges you can use this as an orientation $\mathcal O$. Definition A left-hand turn path on $(\Gamma, \mathcal O)$ is a closed ...
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If $N\lhd H×K$ then $N$ is abelian or $N$ intersects one of $H$ or $K$ nontrivially

I am thinking on this problem: If $N\lhd H×K$ then either $N$ is abelian or $N$ intersects one of $H$ or $K$ nontrivially. I assume; $N$ is not abelian so, there is $(n,n')$ and $(m,m')$ in $N$ ...