(Eilenberg-Moore) Let G be a (possibly nonabelian) group....

(Eilenberg-Moore) Let $G$ be a (possibly nonabelian) group. (i) If $H$ is a proper subgroup of a group $G$, prove that there exist a group $L$ and distinct homomorphisms $f, g: G \rightarrow L$ with $f|H=g| H$. Hint. Define $L=S_{X}$, where $X$ denotes the family of all the left cosets of $H$ in $G$ together with an additional element, denoted $\infty$. If $a \in G$, define $f(a)=f_{a} \in S_{X}$ by $f_{a}(\infty)=$ $\infty$ and $f_{a}(b H)=a b H$. Define $g: G \rightarrow S_{X}$ by $g=\gamma f$, where $\gamma \in S_{X}$ is conjugation by the transposition $(H, \infty)$. (ii) Prove that a homomorphism $\varphi: A \rightarrow G$, where $A$ and $G$ are groups, is surjective if and only if it is an epimorphism in Groups.

(Eilenberg-Moore) Let $G$ be a (possibly nonabelian) group.
(i) If $H$ is a proper subgroup of a group $G$, prove that there exist a group $L$ and distinct homomorphisms $f, g: G \rightarrow L$ with $f|H=g| H$.
Hint. Define $L=S_{X}$, where $X$ denotes the family of all the left cosets of $H$ in $G$ together with an additional element, denoted $\infty$. If $a \in G$, define $f(a)=f_{a} \in S_{X}$ by $f_{a}(\infty)=$ $\infty$ and $f_{a}(b H)=a b H$. Define $g: G \rightarrow S_{X}$ by $g=\gamma f$, where $\gamma \in S_{X}$ is conjugation by the transposition $(H, \infty)$.
(ii) Prove that a homomorphism $\varphi: A \rightarrow G$, where $A$ and $G$ are groups, is surjective if and only if it is an epimorphism in Groups.

Key Concepts

Permutation Groups

A permutation group, often represented as a symmetric group, consists of all the bijections of a set, with the group operation being function composition. Permutation groups provide concrete examples of abstract groups and serve as a way to represent group elements as symmetries of a set. They are particularly useful for visualizing group actions, such as the action on cosets.

Epimorphisms in the Category of Groups

In category theory, an epimorphism is a morphism that is right-cancellative, meaning that if it is composed with two maps yielding the same result, then those maps must be equal. In the category of groups, a homomorphism is an epimorphism if and only if it is surjective. This equivalence is a key concept in algebra, linking the categorical notion of 'epicness' with the familiar concept of a surjective function in group theory.

Cosets

A coset is a subset of a group formed by multiplying all elements of a subgroup by a fixed element from the group. In the case of left cosets, for a subgroup H of a group G and an element a in G, the left coset is given by aH. Cosets partition the group into disjoint subsets and are central in understanding group actions, normal subgroups, and the index of a subgroup.

Group Homomorphisms

A group homomorphism is a function between two groups that preserves the group operation, meaning it satisfies the property f(ab) = f(a)f(b) for any two elements in the domain group. This concept captures the idea of a structure-preserving map, and it plays a central role in studying relations between groups and in constructing new groups from known ones.

Subgroups

A subgroup is a subset of a group that itself forms a group under the same operation. A proper subgroup is one that is strictly smaller than the whole group, meaning it does not contain every element of the larger group. This concept is important for understanding the internal structure of groups and for constructing examples where certain properties differentiate the subgroup from the entire group.

Groups

A group is a set equipped with a binary operation that satisfies the properties of closure, associativity, an identity element, and the existence of inverses for every element. Groups are fundamental algebraic structures that serve as abstractions of symmetry, and they are widely studied in many branches of mathematics and physics.

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