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A subgroup is a smaller group formed within a larger group, maintaining the structure and properties of the original group. Specifically, a subset of a group G is a subgroup if it is itself a group under the same operation defined on G. To be a subgroup, the set must satisfy three conditions:

• Closure: For any elements a and b in the subgroup, the product ab is also in the subgroup.
• Identity: The subgroup contains the identity element of G.
• Inverses: For every element a in the subgroup, its inverse a^-1 is also in the subgroup.

The notation H ≤ G denotes that H is a subgroup of G. For example, the set of all even integers under addition is a subgroup of the set of all integers.

The centralizer of an element in a group is the set of all elements in the group that commute with that element. Formally, the centralizer of an element x in a group G is defined as:

\[C_G(x) = \{ g \in G \; | \; gx = xg \}\]

This means that for every element g in C_G(x), the product of g and x is the same regardless of the order of multiplication. The concept of a centralizer helps to identify the elements that 'commute' with a particular element within the group.

In the context of the given exercise, if H is a subgroup of G, the centralizer of x in H, denoted C_H(x), consists of elements in H that commute with x. The exercise shows how to relate this set to the intersection of H and the centralizer of x in G, written as C_H(x) = H ∩ C_G(x).

A conjugacy class of an element in a group is the set of elements that can be transformed into one another by conjugation. Conjugation involves an element g in the group, and it transforms an element x into gxg^-1.

For a group G and an element x in G, the conjugacy class of x is denoted by x^G, and it is:

\[x^G = \{ gxg^{-1} \; | \; g \in G \}\]

This means all elements that are 'similar' to x by the group operation are in the same conjugacy class. The size of a conjugacy class is an important aspect because it depicts how many unique elements can be formed through conjugation.

In the exercise, the conjugacy class x^H in a subgroup H of index 2 in G has a size that is either equal to or half the size of the conjugacy class x^G in the full group G. This result comes from the limited number of cosets of H in G and how conjugates partition accordingly.

The index of a subgroup H in a group G is the number of distinct cosets of H in G. A coset of H in G is a set formed by multiplying all elements of H by a particular element from G.

Mathematically, the index is defined as:

\[|G : H| = \text{Number of distinct cosets of H in G}\]

For instance, if H has index 2 in G, then there are exactly two cosets: H and G \ H. This situation often arises in the exercise when considering how elements and their properties distribute across the cosets.

In part (ii) of the exercise, we demonstrated that if H is a subgroup of index 2, then a given element x's conjugacy class in H either retains the total size from G or reduces to half in size, reflecting the subgroup's distinct characteristics.