G is a Group with Identity e

First, let's define what a group is. A group is a set G together with a binary operation * that satisfies four conditions: closure, associativity, identity, and inverse. The identity element, denoted by e, is an element in G such that for any element g in G, g * e = e * g = g.

H is an Abelian Non-trivial Proper Subgroup of G

Let's break down the definition of H:
- Abelian: A group is abelian if the operation * is commutative, meaning that for any two elements g and h in G, g * h = h * g.
- Non-trivial: A subgroup is non-trivial if it is not equal to the identity element alone. In other words, H contains at least one element other than e.
- Proper: A subgroup is proper if it is a subset of G but not equal to G itself.

H ∩ gHg⁻¹ = {e} for all g ∉ H

This property states that the intersection of H and gHg⁻¹ is equal to the identity element e for all elements g that do not belong to H. Let's break it down further:
- H ∩ gHg⁻¹ is the set of elements that are common to both H and gHg⁻¹. This means that for any element h in H and any element g in G that is not in H, ghg⁻¹ is not in H.
- {e} is the set containing only the identity element e.

k = {g ∈ G : gh = hg for all h ∈ H}

Now, let's define k as the set of elements g in G such that for any element h in H, gh = hg. This means that k is the set of elements that commute with all elements of H.

Summary

In summary, we have a group G with identity e. We also have an abelian non-trivial proper subgroup H of G. The property H ∩ gHg⁻¹ = {e} for all g ∉ H states that the intersection of H and gHg⁻¹ is equal to the identity element e for all elements g that do not belong to H. Additionally, we define k as the set of elements g in G that commute with all elements of H.