Let G be a group and suppose that H is a subgroup of G with...

Let $G$ be a group and suppose that $H$ is a subgroup of $G$ with the property that for any $a$ in $G$ we have $a H=H a$. (That is, every element of the form $a h$ where $h$ is some element of $H$ can be written in the form $h_{1} a$ for some $h_{1} \in H .$ ) If $a$ has order 2, prove that the set $K=H \cup a H$ is a subgroup of $G$. Generalize to the case that $|a|=k$.

Key Concepts

Group and Subgroup

A group is a set combined with an operation that satisfies closure, associativity, has an identity element, and every element has an inverse. A subgroup is a subset of a group that itself satisfies these properties. Recognizing subgroup structure is essential when analyzing group behavior and constructing new subgroups.

Cosets

Cosets are subsets formed by multiplying a subgroup by an element from the group. A left coset has the form aH and a right coset Ha. Cosets are used to partition groups and are critical in understanding the structure and relationships between a subgroup and the full group, particularly in the context of Lagrange’s theorem.

Normal Subgroups

A subgroup is normal if it is invariant under conjugation by any element of the group, meaning aH = Ha for every a in the group. Normal subgroups are foundational because they allow for the formation of quotient groups and simplify the analysis of group extensions and subgroup unions.

Order of an Element

The order of an element in a group is the smallest positive integer k such that raising the element to the k-th power returns the identity element. This concept is pivotal in understanding the cyclic behavior of elements and is used to analyze subgroup structures built from elements of finite order.

Subgroup Criterion

The subgroup criterion provides a method to verify that a subset of a group is indeed a subgroup by ensuring the subset is closed under the group operation and taking inverses. This is particularly important when proving that a union of cosets forms a subgroup.

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