Let G be a group and H and K are subgroups of G Show H K is

Let G be a group and H and K are subgroups of G. Show H K is also a subgroup of G. In addition if H and K are normal subgroups, then H K is also a normal subgroup of G.

Solution

ANSWER:-

Assume that H and K are subgroups of G. using the One-Step Subgroup Test. Let x, y H K. Then x, y H and x, y K. Since both H and K are groups, we have xy^-1 H and xy^-1 K. This implies that xy^-1 H K.   Thus H K is a subgroup of G

If H and K are normal subgroups,then H K is also a subgroup of G.

proof :- The intersection of two subgroups is again a subgroup, so H K is a subgroup of G.

Let xH K and g G. Then g^-1xg H (as xH is a subgroup of G) and similarly g^-1xg K. Hence g^-1xg H K and so H K is a subgroup of G.