Question

In: Math

1. Let N be a normal subgroup of G and let H be any subgroup of...

1. Let N be a normal subgroup of G and let H be any subgroup of G. Let HN = {hn|h ∈ H,n ∈

N}. Show that HN is a subgroup of G, and is the smallest subgroup containing both N and H.

Expert Solution

If H is a subgroup of N and if N is a normal subgroup. Then we have to prove HN is also a subgroup.

Now

Now consider an element and where and .

Now if as we know from subgroup testthen the HN is a subgroup.

Now

Now as N is a subgroup. Let the element be

Hence

Now

Or,

Now, as N is normal subgroup. Let the element be

Or,

Now as H is a subgroup

so

So HN is a subgroup.

(B) Now consider G is any group that contains H and N. Now hence it must contain every element of the form As G is closed under multiplication. Hence . Hence every group that contains H and N must contain HN. Hence HN is the smallest group that contains both H and N

milcah answered 2 years ago

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