Paul wants to gift wrap the boxes. He has 1,000.00 square inches of wrapping paper. Does Paul have enough to wrap all three boxe
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Suppose first that is a normal subgroup. Then by definition we must have for all
,
for every
. Let
and choose
(
). By hypothesis we have
, i.e.
for some
, thus
. So we have
. You can prove
in the same way.
Suppose for all
. Let
, we have to prove
for every
. So, let
. We have that
for some
(by the hypothesis). hence we have
. Because
was chosen arbitrarily we have the desired
.