MATH 451 FIRST MID-TERM
NAME: John Q. Public
2 MATH 451 FIRST MID-TERM
Question 1. Let Hbe a nonempty subset of the group G. Prove that His a
subgroup of Giff a b−1∈Hfor all a,b∈H.
First suppose that His a subgroup of G. Then His closed under multiplication
and taking inverses. Hence if a,b∈H, then b−1∈Hand so ab−1∈H.
Next suppose that ∅ 6=H⊆Gis such that ab−1∈Hfor all a,b∈H. Since
H6=∅, there exists an element a∈Hand hence 1 = aa−1∈H. It follows that if
a∈H, then a−1= 1 a−1∈H. Finally suppose that a,b∈H. Then b−1∈Hand
so ab =a(b−1)−1∈H. Thus His a subgroup of G.