Category: Group Theory

A Group Homomorphism that Factors though Another Group

Problem 490

Let $G, H, K$ be groups. Let $f:G\to K$ be a group homomorphism and let $\pi:G\to H$ be a surjective group homomorphism such that the kernel of $\pi$ is included in the kernel of $f$: $\ker(\pi) \subset \ker(f)$.

Define a map $\tilde{f}:H\to K$ as follows.
For each $h\in H$, there exists $g\in G$ such that $\pi(g)=h$ since $\pi:G\to H$ is surjective.
Define $\tilde{f}:H\to K$ by $\tilde{f}(h)=f(g)$.

(a) Prove that the map $\tilde{f}:H\to K$ is well-defined.

(b) Prove that $\tilde{f}:H\to K$ is a group homomorphism.

 
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The Order of a Conjugacy Class Divides the Order of the Group

Problem 455

Let $G$ be a finite group.
The centralizer of an element $a$ of $G$ is defined to be
\[C_G(a)=\{g\in G \mid ga=ag\}.\]

A conjugacy class is a set of the form
\[\Cl(a)=\{bab^{-1} \mid b\in G\}\] for some $a\in G$.


(a) Prove that the centralizer of an element of $a$ in $G$ is a subgroup of the group $G$.

(b) Prove that the order (the number of elements) of every conjugacy class in $G$ divides the order of the group $G$.

 
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Injective Group Homomorphism that does not have Inverse Homomorphism

Problem 443

Let $A=B=\Z$ be the additive group of integers.
Define a map $\phi: A\to B$ by sending $n$ to $2n$ for any integer $n\in A$.

(a) Prove that $\phi$ is a group homomorphism.

(b) Prove that $\phi$ is injective.

(c) Prove that there does not exist a group homomorphism $\psi:B \to A$ such that $\psi \circ \phi=\id_A$.

 
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Fundamental Theorem of Finitely Generated Abelian Groups and its application

Problem 420

In this post, we study the Fundamental Theorem of Finitely Generated Abelian Groups, and as an application we solve the following problem.

Problem.
Let $G$ be a finite abelian group of order $n$.
If $n$ is the product of distinct prime numbers, then prove that $G$ is isomorphic to the cyclic group $Z_n=\Zmod{n}$ of order $n$.

 
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