(Baer). Call a group E injective ^3 in Groups if it solves...

(Baer). Call a group E injective $^{3}$ in Groups if it solves the obvious universal mapping problem: for every group $G$ and every (not necessarily abelian) subgroup $S \subseteq G$, every homomorphism $f^{\prime}: S \rightarrow E$ can be extended to $G$ : Prove that $E$ is injective if and only if $E=\{1\}$. Hint. Let $A$ be free with basis $\{x, y\}$, and let $B$ be the semidirect product $B=A \rtimes\langle z\rangle$, where $z$ is an element of order 2 that acts on A by $z x z=y$ and $z y z=x$.

(Baer). Call a group E injective $^{3}$ in Groups if it solves the obvious universal mapping problem: for every group $G$ and every (not necessarily abelian) subgroup $S \subseteq G$, every homomorphism $f^{\prime}: S \rightarrow E$ can be extended to $G$ :
Prove that $E$ is injective if and only if $E=\{1\}$.
Hint. Let $A$ be free with basis $\{x, y\}$, and let $B$ be the semidirect product $B=A \rtimes\langle z\rangle$, where $z$ is an element of order 2 that acts on A by $z x z=y$ and $z y z=x$.

Key Concepts

Injective Object

An injective object in category theory is one that satisfies an extension property: for every monomorphism from an object S into another object G and any morphism from S to the injective object, there exists an extension of that morphism to all of G. In the context of groups, an injective group would allow every homomorphism defined on a subgroup to be extended to the whole group.

Universal Mapping Property

The universal mapping property refers to the capability of extending morphisms from a substructure to the entirety of a structure. Specifically, for groups, it means that for any subgroup S of a group G and any homomorphism from S to a target group E, there exists an extended homomorphism from G to E. This property is key to understanding injectivity in the category of groups.

Free Group

A free group is a group with a freely generating set, meaning there are no relations among the generators aside from the group axioms. Free groups serve as fundamental examples in group theory, providing a setting in which abstract properties such as extending homomorphisms can be effectively studied.

Semi-direct Product

A semi-direct product is a construction in group theory that forms a new group from two groups, where one group acts on the other by automorphisms. This construction is used to create examples with nontrivial structural properties, which are particularly useful in testing the boundaries of universal mapping properties and injectivity.

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