Question

(a) Let H and K be subgroups of a group G. Prove that$\mathrm{H}\cap \mathrm{K}$ is a subgroup of G.

Short answer

It is proved that if H and K are subgroups of group G, then their intersection $\mathrm{H}\cap \mathrm{K}$is a subgroup of G.

Step-by-step solution

Subgroups of a group G

We have thedefinition of subgroupthat,

Let (G,*) be a group under binary operation, say *-A non-empty subset H of G is said to be a subgroup of G, if (H,*) is itself a group.

Suppose H and K are two subgroups of a group G.

To show that H∩K is a subgroup of G.

Let and be two subgroups of a group .

Now, $\mathrm{e}\in \mathrm{H}$and$\mathrm{e}\in \mathrm{K}$

$$\begin{gathered} \Rightarrow \mathrm{e}\in \mathrm{H}\cap \mathrm{K} \\ \Rightarrow \mathrm{H}\cap \mathrm{K}\ne \mathrm{\varphi }. \\ \mathrm{Let}\mathrm{a},\mathrm{b}\in \mathrm{H}\cap \mathrm{K} \\ \mathrm{a},\mathrm{b}\in \mathrm{H}\mathrm{and}\mathrm{a},\mathrm{b}\in \mathrm{K} \\ {\mathrm{ab}}^{-1}\in \mathrm{H}\mathrm{and}{\mathrm{ab}}^{-1}\in \mathrm{K}...(\mathrm{therefore},\mathrm{H},\mathrm{K}\mathrm{are}\mathrm{subgroups}) \\ {\mathrm{ab}}^{-1}\in \mathrm{H}\cap \mathrm{K} \end{gathered}$$

Hence $\mathrm{H}\cap \mathrm{K}$, is a subgroup of G.