Tagged: finite group

If a Half of a Group are Elements of Order 2, then the Rest form an Abelian Normal Subgroup of Odd Order

Problem 575

Let $G$ be a finite group of order $2n$.
Suppose that exactly a half of $G$ consists of elements of order $2$ and the rest forms a subgroup.
Namely, suppose that $G=S\sqcup H$, where $S$ is the set of all elements of order in $G$, and $H$ is a subgroup of $G$. The cardinalities of $S$ and $H$ are both $n$.

Then prove that $H$ is an abelian normal subgroup of odd order.

 

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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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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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Generators of the Augmentation Ideal in a Group Ring

Problem 302

Let $R$ be a commutative ring with $1$ and let $G$ be a finite group with identity element $e$. Let $RG$ be the group ring. Then the map $\epsilon: RG \to R$ defined by
\[\epsilon(\sum_{i=1}^na_i g_i)=\sum_{i=1}^na_i,\] where $a_i\in R$ and $G=\{g_i\}_{i=1}^n$, is a ring homomorphism, called the augmentation map and the kernel of $\epsilon$ is called the augmentation ideal.

(a) Prove that the augmentation ideal in the group ring $RG$ is generated by $\{g-e \mid g\in G\}$.

(b) Prove that if $G=\langle g\rangle$ is a finite cyclic group generated by $g$, then the augmentation ideal is generated by $g-e$.
 

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