# mmc%20rep%20of%20mcg

mmc%20rep%20of%20mcg

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• Equivalent Definitions of Characteristic Subgroups. Center is Characteristic. Let $H$ be a subgroup of a group $G$. We call $H$ characteristic in $G$ if for any automorphism $\sigma\in \Aut(G)$ of $G$, we have $\sigma(H)=H$. (a) Prove that if $\sigma(H) \subset H$ for all $\sigma \in \Aut(G)$, then $H$ is characteristic in $G$. (b) Prove that the center […]
• Dot Product, Lengths, and Distances of Complex Vectors For this problem, use the complex vectors $\mathbf{w}_1 = \begin{bmatrix} 1 + i \\ 1 - i \\ 0 \end{bmatrix} , \, \mathbf{w}_2 = \begin{bmatrix} -i \\ 0 \\ 2 - i \end{bmatrix} , \, \mathbf{w}_3 = \begin{bmatrix} 2+i \\ 1 - 3i \\ 2i \end{bmatrix} .$ Suppose $\mathbf{w}_4$ is […]
• Two Matrices are Nonsingular if and only if the Product is Nonsingular An $n\times n$ matrix $A$ is called nonsingular if the only vector $\mathbf{x}\in \R^n$ satisfying the equation $A\mathbf{x}=\mathbf{0}$ is $\mathbf{x}=\mathbf{0}$. Using the definition of a nonsingular matrix, prove the following statements. (a) If $A$ and $B$ are $n\times […] • Is the Set of Nilpotent Element an Ideal? Is it true that a set of nilpotent elements in a ring$R$is an ideal of$R$? If so, prove it. Otherwise give a counterexample. Proof. We give a counterexample. Let$R$be the noncommutative ring of$2\times 2$matrices with real […] • The 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 […] • Every Group of Order 12 Has a Normal Subgroup of Order 3 or 4 Let$G$be a group of order$12$. Prove that$G$has a normal subgroup of order$3$or$4$. Hint. Use Sylow's theorem. (See Sylow’s Theorem (Summary) for a review of Sylow's theorem.) Recall that if there is a unique Sylow$p$-subgroup in a group$GH$, then it is […] • Show that the Given 2 by 2 Matrix is Singular Consider the matrix$M = \begin{bmatrix} 1 & 4 \\ 3 & 12 \end{bmatrix}$. (a) Show that$M$is singular. (b) Find a non-zero vector$\mathbf{v}$such that$M \mathbf{v} = \mathbf{0}$, where$\mathbf{0}$is the$2$-dimensional zero vector. Solution. (a) Show […] • Range, Null Space, Rank, and Nullity of a Linear Transformation from$\R^2$to$\R^3$Define the map$T:\R^2 \to \R^3$by$T \left ( \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}\right )=\begin{bmatrix} x_1-x_2 \\ x_1+x_2 \\ x_2 \end{bmatrix}$. (a) Show that$T$is a linear transformation. (b) Find a matrix$A\$ such that […]