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  • If the Order is an Even Perfect Number, then a Group is not SimpleIf the Order is an Even Perfect Number, then a Group is not Simple (a) Show that if a group $G$ has the following order, then it is not simple. $28$ $496$ $8128$ (b) Show that if the order of a group $G$ is equal to an even perfect number then the group is not simple. Hint. Use Sylow's theorem. (See the post Sylow’s Theorem […]
  • Conjugate of the Centralizer of a Set is the Centralizer of the Conjugate of the SetConjugate of the Centralizer of a Set is the Centralizer of the Conjugate of the Set Let $X$ be a subset of a group $G$. Let $C_G(X)$ be the centralizer subgroup of $X$ in $G$. For any $g \in G$, show that $gC_G(X)g^{-1}=C_G(gXg^{-1})$.   Proof. $(\subset)$ We first show that $gC_G(X)g^{-1} \subset C_G(gXg^{-1})$. Take any $h\in C_G(X)$. Then for […]
  • Elementary Questions about a MatrixElementary Questions about a Matrix Let \[A=\begin{bmatrix} -5 & 0 & 1 & 2 \\ 3 &8 & -3 & 7 \\ 0 & 11 & 13 & 28 \end{bmatrix}.\] (a) What is the size of the matrix $A$? (b) What is the third column of $A$? (c) Let $a_{ij}$ be the $(i,j)$-entry of $A$. Calculate $a_{23}-a_{31}$. […]
  • Group Homomorphism from $\Z/n\Z$ to $\Z/m\Z$ When $m$ Divides $n$Group Homomorphism from $\Z/n\Z$ to $\Z/m\Z$ When $m$ Divides $n$ Let $m$ and $n$ be positive integers such that $m \mid n$. (a) Prove that the map $\phi:\Zmod{n} \to \Zmod{m}$ sending $a+n\Z$ to $a+m\Z$ for any $a\in \Z$ is well-defined. (b) Prove that $\phi$ is a group homomorphism. (c) Prove that $\phi$ is surjective. (d) Determine […]
  • The Preimage of Prime ideals are Prime IdealsThe Preimage of Prime ideals are Prime Ideals Let $f: R\to R'$ be a ring homomorphism. Let $P$ be a prime ideal of the ring $R'$. Prove that the preimage $f^{-1}(P)$ is a prime ideal of $R$.   Proof. The preimage of an ideal by a ring homomorphism is an ideal. (See the post "The inverse image of an ideal by […]
  • Non-Example of a Subspace in 3-dimensional Vector Space $\R^3$Non-Example of a Subspace in 3-dimensional Vector Space $\R^3$ Let $S$ be the following subset of the 3-dimensional vector space $\R^3$. \[S=\left\{ \mathbf{x}\in \R^3 \quad \middle| \quad \mathbf{x}=\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}, x_1, x_2, x_3 \in \Z \right\}, \] where $\Z$ is the set of all integers. […]
  • The Quotient by the Kernel Induces an Injective HomomorphismThe Quotient by the Kernel Induces an Injective Homomorphism Let $G$ and $G'$ be a group and let $\phi:G \to G'$ be a group homomorphism.  Show that $\phi$ induces an injective homomorphism from $G/\ker{\phi} \to G'$.   Outline. Define $\tilde{\phi}([g])=\phi(g)$ and show that this is well-defined. Show […]
  • Order of the Product of Two Elements in an Abelian GroupOrder of the Product of Two Elements in an Abelian Group Let $G$ be an abelian group with the identity element $1$. Let $a, b$ be elements of $G$ with order $m$ and $n$, respectively. If $m$ and $n$ are relatively prime, then show that the order of the element $ab$ is $mn$.   Proof. Let $r$ be the order of the element […]

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