Kyushu-University-Linear-Algebra

• Even Perfect Numbers and Mersenne Prime Numbers Prove that if $2^n-1$ is a Mersenne prime number, then $N=2^{n-1}(2^n-1)$ is a perfect number. On the other hand, prove that every even perfect number $N$ can be written as $N=2^{n-1}(2^n-1)$ for some Mersenne prime number $2^n-1$.   Definitions. In this post, a […]
• A Simple Abelian Group if and only if the Order is a Prime Number Let $G$ be a group. (Do not assume that $G$ is a finite group.) Prove that $G$ is a simple abelian group if and only if the order of $G$ is a prime number.   Definition. A group $G$ is called simple if $G$ is a nontrivial group and the only normal subgroups of $G$ is […]
• Any Finite Group Has a Composition Series Let $G$ be a finite group. Then show that $G$ has a composition series.   Proof. We prove the statement by induction on the order $|G|=n$ of the finite group. When $n=1$, this is trivial. Suppose that any finite group of order less than $n$ has a composition […]
• A Homomorphism from the Additive Group of Integers to Itself Let $\Z$ be the additive group of integers. Let $f: \Z \to \Z$ be a group homomorphism. Then show that there exists an integer $a$ such that $f(n)=an$ for any integer $n$.   Hint. Let us first recall the definition of a group homomorphism. A group homomorphism from a […]
• The Center of a p-Group is Not Trivial Let $G$ be a group of order $|G|=p^n$ for some $n \in \N$. (Such a group is called a $p$-group.) Show that the center $Z(G)$ of the group $G$ is not trivial. Hint. Use the class equation. Proof. If $G=Z(G)$, then the statement is true. So suppose that $G\neq […] • The Sum of Cosine Squared in an Inner Product Space Let$\mathbf{v}$be a vector in an inner product space$V$over$\R$. Suppose that$\{\mathbf{u}_1, \dots, \mathbf{u}_n\}$is an orthonormal basis of$V$. Let$\theta_i$be the angle between$\mathbf{v}$and$\mathbf{u}_i$for$i=1,\dots, n$. Prove that \[\cos […] • 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, […]
• Two Normal Subgroups Intersecting Trivially Commute Each Other Let $G$ be a group. Assume that $H$ and $K$ are both normal subgroups of $G$ and $H \cap K=1$. Then for any elements $h \in H$ and $k\in K$, show that $hk=kh$.   Proof. It suffices to show that $h^{-1}k^{-1}hk \in H \cap K$. In fact, if this it true then we have […]