# Purdue-Algebra-Exam-eye-catch

by Yu ·

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- Example of an Infinite Group Whose Elements Have Finite Orders Is it possible that each element of an infinite group has a finite order? If so, give an example. Otherwise, prove the non-existence of such a group. Solution. We give an example of a group of infinite order each of whose elements has a finite order. Consider […]
- True of False Problems on Determinants and Invertible Matrices Determine whether each of the following statements is True or False. (a) If $A$ and $B$ are $n \times n$ matrices, and $P$ is an invertible $n \times n$ matrix such that $A=PBP^{-1}$, then $\det(A)=\det(B)$. (b) If the characteristic polynomial of an $n \times n$ matrix $A$ […]
- Is the Set of All Orthogonal Matrices a Vector Space? An $n\times n$ matrix $A$ is called orthogonal if $A^{\trans}A=I$. Let $V$ be the vector space of all real $2\times 2$ matrices. Consider the subset \[W:=\{A\in V \mid \text{$A$ is an orthogonal matrix}\}.\] Prove or disprove that $W$ is a subspace of […]
- If Every Nonidentity Element of a Group has Order 2, then it’s an Abelian Group Let $G$ be a group. Suppose that the order of nonidentity element of $G$ is $2$. Then show that $G$ is an abelian group. Proof. Let $x$ and $y$ be elements of $G$. Then we have \[1=(xy)^2=(xy)(xy).\] Multiplying the equality by $yx$ from the right, we […]
- The Formula for the Inverse Matrix of $I+A$ for a $2\times 2$ Singular Matrix $A$ Let $A$ be a singular $2\times 2$ matrix such that $\tr(A)\neq -1$ and let $I$ be the $2\times 2$ identity matrix. Then prove that the inverse matrix of the matrix $I+A$ is given by the following formula: \[(I+A)^{-1}=I-\frac{1}{1+\tr(A)}A.\] Using the formula, calculate […]
- Finite Group and Subgroup Criteria Let $G$ be a finite group and let $H$ be a subset of $G$ such that for any $a,b \in H$, $ab\in H$. Then show that $H$ is a subgroup of $G$. Proof. Let $a \in H$. To show that $H$ is a subgroup of $G$, it suffices to show that the inverse $a^{-1}$ is in $H$. If […]
- Linearly Independent vectors $\mathbf{v}_1, \mathbf{v}_2$ and Linearly Independent Vectors $A\mathbf{v}_1, A\mathbf{v}_2$ for a Nonsingular Matrix Let $\mathbf{v}_1$ and $\mathbf{v}_2$ be $2$-dimensional vectors and let $A$ be a $2\times 2$ matrix. (a) Show that if $\mathbf{v}_1, \mathbf{v}_2$ are linearly dependent vectors, then the vectors $A\mathbf{v}_1, A\mathbf{v}_2$ are also linearly dependent. (b) If $\mathbf{v}_1, […]
- Subgroup Containing All $p$-Sylow Subgroups of a Group Suppose that $G$ is a finite group of order $p^an$, where $p$ is a prime number and $p$ does not divide $n$. Let $N$ be a normal subgroup of $G$ such that the index $|G: N|$ is relatively prime to $p$. Then show that $N$ contains all $p$-Sylow subgroups of […]