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• 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 […] • Group of Order pq is Either Abelian or the Center is Trivial Let G be a group of order |G|=pq, where p and q are (not necessarily distinct) prime numbers. Then show that G is either abelian group or the center Z(G)=1. Hint. Use the result of the problem "If the Quotient by the Center is Cyclic, then the Group is […] • The Symmetric Group is a Semi-Direct Product of the Alternating Group and a Subgroup \langle(1,2) \rangle Prove that the symmetric group S_n, n\geq 3 is a semi-direct product of the alternating group A_n and the subgroup \langle(1,2) \rangle generated by the element (1,2). Definition (Semi-Direct Product). Internal Semi-Direct-Product Recall that a group G is […] • Possibilities For the Number of Solutions for a Linear System Determine whether the following systems of equations (or matrix equations) described below has no solution, one unique solution or infinitely many solutions and justify your answer. (a) \[\left\{ \begin{array}{c} ax+by=c \\ dx+ey=f, \end{array} \right.$ where $a,b,c, d$ […]
• Irreducible Polynomial Over the Ring of Polynomials Over Integral Domain Let $R$ be an integral domain and let $S=R[t]$ be the polynomial ring in $t$ over $R$. Let $n$ be a positive integer. Prove that the polynomial $f(x)=x^n-t$ in the ring $S[x]$ is irreducible in $S[x]$.   Proof. Consider the principal ideal $(t)$ generated by $t$ […]
• A Group Homomorphism is Injective if and only if Monic Let $f:G\to G'$ be a group homomorphism. We say that $f$ is monic whenever we have $fg_1=fg_2$, where $g_1:K\to G$ and $g_2:K \to G$ are group homomorphisms for some group $K$, we have $g_1=g_2$. Then prove that a group homomorphism $f: G \to G'$ is injective if and only if it is […]
• The Center of the Symmetric group is Trivial if $n>2$ Show that the center $Z(S_n)$ of the symmetric group with $n \geq 3$ is trivial. Steps/Hint Assume $Z(S_n)$ has a non-identity element $\sigma$. Then there exist numbers $i$ and $j$, $i\neq j$, such that $\sigma(i)=j$ Since $n\geq 3$ there exists another […]
• Non-Abelian Group of Order $pq$ and its Sylow Subgroups Let $G$ be a non-abelian group of order $pq$, where $p, q$ are prime numbers satisfying $q \equiv 1 \pmod p$. Prove that a $q$-Sylow subgroup of $G$ is normal and the number of $p$-Sylow subgroups are $q$.   Hint. Use Sylow's theorem. To review Sylow's theorem, check […]