TIFR-2012-Maths-A: 1

Question

True/False Question:

If  $H_{1}$ & $H_{2}$ are subgroups of a group $G$ then $H_{1} .H_{2}=\left \{ h_{1} h_{2}\in G \mid h_{1}\in H_{1},h_{2}\in H_{2}\right \}$ is a subgroup of $G$.

Answer 1

As $H_1.H_2 \subseteq G$, now we only have to check if it satisfies the properties of group

Now take $ h_1 = e, \text{ then } H_2 \subseteq H_1.H_2, \text{ same way take } h_2 = e$  then $H_1 \subseteq H_1.H_2$

Checking closure property i.e $x.y ∈ H_1.H_2, \text{for all x, y} ∈ H_1.H_2$

Suppose $a \in H_1 \text{ and } b ∈ H_2, \text{ then } a.b \in H_1.H_2 \text{, but it is not necessary that } b.a \in H_1.H_2$, unless the group G satisfies commutative property or $H_1 \subset H_2$ or vice versa.

$\therefore$ Closure property is not satsfied, statement is False.