If $K$ is subgroup of $H$ and $H$ is a subgroup of $G$, then $K$ is a subgroup of $G$

Question

Question
1. Let $K\subseteq H\subseteq G$ and if $K$ is subgroup of $H$ and $H$ is a subgroup of $G$, then $K$ is a subgroup of $G$.
2. Let $K\subseteq H\subseteq G$ and if $K$ is a subgroup of $G$ and $H$ is a subgroup of $G$, then $K$ is a subgroup of $H$.

Solution
1. Clearly, $K\subseteq G$. Let $p,q\in K$. How can I show that $pq^{-1}\in K$, to prove that $K$ is a subgroup of $G$.
Also how can I prove the second case?

How is "subgroup" actually defined ? And what criteria for testing this do you have available? — Hagen von Eitzen, May 10 '15 at 13:31

score 1 · Accepted Answer · answered May 10 '15 at 14:09

a- $pq^{-1}\in K$ because $K$ is a subgroup of $H$. Therefore $pq^{-1}\in K$ as a subset of $G$ and $K$ is non empty. So $K$ is a subgroup of $G$

b- $K$ is non empty because it contains $e$ as a subgroup of $G$. For two elements $p$ and $q$ of $K$ we have $pq^{-1}\in K$ because $K$ is s subgroup of $G$ and therefore as a subset of $H$ it is a subgroup of $H$.

score 0 · Answer 2 · answered May 10 '15 at 14:05

do not be over-rigid in your approach. the $pq^{-1} \in G$ criterion is clumsy for a question of this type.

what you need is: $$ A \le B \Leftarrow \Rightarrow (\text{A is a group}) \land (A \subseteq B) \tag{1} $$ i.e. A is a subgroup of B iff A is a group and A is a subset of B

solve your problem using this principle, then, if you wish, prove (1) using your preferred test

If $K$ is subgroup of $H$ and $H$ is a subgroup of $G$, then $K$ is a subgroup of $G$

2 Answers2