Let G H1 times H2 Let N be a normal subgroup of G such that

Let G = H_1 times H_2. Let N be a normal subgroup of G such that N H_1 = N H_2 = {e}. Prove that N lies in the center of G.

Solution

solution-:Suppose that N lies in the center

We know that e 2 H1 and e 2 H2, since H1 and H2 are
subgroups, so e 2 H1 \\ H2.
Next let h 2 H1 \\ H2. Then h 2 H1 and H1 is a subgroup, so
h1 2 H1. Similarly, h 2 H2 and H2 is a subgroup, so h1 2 H2. This
shows that h1 2 H1 \\ H2.
Finally let h, h 2 H1 \\ H2. Then h 2 H1 and h 2 H1 and H1 is
a subgroup, so hh 2 H1. Similarly h 2 H2 and h 2 H2 and H2 is a
subgroup, so hh 2 H2. This shows that hh 2 H1 \\ H2.
We have shown that H1\\H2 contains the identity, contains inverses,
and is closed under group multiplication. Hence it is a subgroup.Hence it is true