# perfect-numbers

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- Number Theoretical Problem Proved by Group Theory. $a^{2^n}+b^{2^n}\equiv 0 \pmod{p}$ Implies $2^{n+1}|p-1$. Let $a, b$ be relatively prime integers and let $p$ be a prime number. Suppose that we have \[a^{2^n}+b^{2^n}\equiv 0 \pmod{p}\] for some positive integer $n$. Then prove that $2^{n+1}$ divides $p-1$. Proof. Since $a$ and $b$ are relatively prime, at least one […]
- 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 […]
- Transpose of a Matrix and Eigenvalues and Related Questions Let $A$ be an $n \times n$ real matrix. Prove the followings. (a) The matrix $AA^{\trans}$ is a symmetric matrix. (b) The set of eigenvalues of $A$ and the set of eigenvalues of $A^{\trans}$ are equal. (c) The matrix $AA^{\trans}$ is non-negative definite. (An $n\times n$ […]
- Intersection of Two Null Spaces is Contained in Null Space of Sum of Two Matrices Let $A$ and $B$ be $n\times n$ matrices. Then prove that \[\calN(A)\cap \calN(B) \subset \calN(A+B),\] where $\calN(A)$ is the null space (kernel) of the matrix $A$. Definition. Recall that the null space (or kernel) of an $n \times n$ matrix […]
- Determine Conditions on Scalars so that the Set of Vectors is Linearly Dependent Determine conditions on the scalars $a, b$ so that the following set $S$ of vectors is linearly dependent. \begin{align*} S=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}, \end{align*} where \[\mathbf{v}_1=\begin{bmatrix} 1 \\ 3 \\ 1 \end{bmatrix}, […]
- Two Quotients Groups are Abelian then Intersection Quotient is Abelian Let $K, N$ be normal subgroups of a group $G$. Suppose that the quotient groups $G/K$ and $G/N$ are both abelian groups. Then show that the group \[G/(K \cap N)\] is also an abelian group. Hint. We use the following fact to prove the problem. Lemma: For a […]
- Three Linearly Independent Vectors in $\R^3$ Form a Basis. Three Vectors Spanning $\R^3$ Form a Basis. Let $B=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ be a set of three-dimensional vectors in $\R^3$. (a) Prove that if the set $B$ is linearly independent, then $B$ is a basis of the vector space $\R^3$. (b) Prove that if the set $B$ spans $\R^3$, then $B$ is a basis of […]
- Find All the Eigenvalues of 4 by 4 Matrix Find all the eigenvalues of the matrix \[A=\begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 &0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \end{bmatrix}.\] (The Ohio State University, Linear Algebra Final Exam Problem) Solution. We compute the […]