Let H be a subgroup of G If gG show that gHg1g1hghH is also

Let H be a subgroup of G. If g?G, show that gHg?1={g?1hg:h?H} is also a subgroup of G.

Let H be a subgroup of G.lf g G. show that gHg-1\' = also a subgroup of G.

Solution

Suppose that H is normal in G.

Then for all a G, aHa1 H.

Taking a = g and a = g 1 we get gHg1 H and g 1Hg H.

Multiplying the second inclusion on the left by g and on the right by g 1 we get, H gHg1 .

so gHg1={g1hg:hH} is also a subgroup of G.