Algebra I – autumn 2024 – Homework 6
Let (G, ·) be a group and n arbitrary positive integer.
1. Prove that if
Hn = {g ∈ G; gn
= 1G}
is a subgroup of G, then H is a normal subgroup.
2. Give an example of a group G such that H is not a subgroup, and explain why it is not a
subgroup.
3. Give an example of a noncommutative group G such that H3 is a nontrivial subgroup, and prove
that H3 is a subgroup that is normal.