Group Homomorphism Sends the Inverse Element to the Inverse Element

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• A Group Homomorphism and an Abelian Group Let $G$ be a group. Define a map $f:G \to G$ by sending each element $g \in G$ to its inverse $g^{-1} \in G$. Show that $G$ is an abelian group if and only if the map $f: G\to G$ is a group homomorphism.   Proof. $(\implies)$ If $G$ is an abelian group, then $f$ […]
• Pullback Group of Two Group Homomorphisms into a Group Let $G_1, G_1$, and $H$ be groups. Let $f_1: G_1 \to H$ and $f_2: G_2 \to H$ be group homomorphisms. Define the subset $M$ of $G_1 \times G_2$ to be \[M=\{(a_1, a_2) \in G_1\times G_2 \mid f_1(a_1)=f_2(a_2)\}.\] Prove that $M$ is a subgroup of $G_1 \times G_2$.   […]
• Basic Properties of Characteristic Groups Definition (automorphism). An isomorphism from a group $G$ to itself is called an automorphism of $G$. The set of all automorphism is denoted by $\Aut(G)$. Definition (characteristic subgroup). A subgroup $H$ of a group $G$ is called characteristic in $G$ if for any $\phi […]
• Cyclic Group if and only if There Exists a Surjective Group Homomorphism From $\Z$ Show that a group $G$ is cyclic if and only if there exists a surjective group homomorphism from the additive group $\Z$ of integers to the group $G$.   Proof. $(\implies)$: If $G$ is cyclic, then there exists a surjective homomorhpism from $\Z$ Suppose that $G$ is […]
• A Group is Abelian if and only if Squaring is a Group Homomorphism Let $G$ be a group and define a map $f:G\to G$ by $f(a)=a^2$ for each $a\in G$. Then prove that $G$ is an abelian group if and only if the map $f$ is a group homomorphism.   Proof. $(\implies)$ If $G$ is an abelian group, then $f$ is a homomorphism. Suppose that […]
• A Homomorphism from the Additive Group of Integers to Itself Let $\Z$ be the additive group of integers. Let $f: \Z \to \Z$ be a group homomorphism. Then show that there exists an integer $a$ such that \[f(n)=an\] for any integer $n$.   Hint. Let us first recall the definition of a group homomorphism. A group homomorphism from a […]
• Abelian Groups and Surjective Group Homomorphism Let $G, G'$ be groups. Suppose that we have a surjective group homomorphism $f:G\to G'$. Show that if $G$ is an abelian group, then so is $G'$.   Definitions. Recall the relevant definitions. A group homomorphism $f:G\to G'$ is a map from $G$ to $G'$ […]
• A Group Homomorphism is Injective if and only if the Kernel is Trivial Let $G$ and $H$ be groups and let $f:G \to K$ be a group homomorphism. Prove that the homomorphism $f$ is injective if and only if the kernel is trivial, that is, $\ker(f)=\{e\}$, where $e$ is the identity element of $G$.     Definitions/Hint. We recall several […]