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  • Any Subgroup of Index 2 in a Finite Group is NormalAny Subgroup of Index 2 in a Finite Group is Normal Show that any subgroup of index $2$ in a group is a normal subgroup. Hint. Left (right) cosets partition the group into disjoint sets. Consider both left and right cosets. Proof. Let $H$ be a subgroup of index $2$ in a group $G$. Let $e \in G$ be the identity […]
  • Non-Abelian Group of Order $pq$ and its Sylow SubgroupsNon-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 […]
  • Common Eigenvector of Two Matrices and Determinant of CommutatorCommon Eigenvector of Two Matrices and Determinant of Commutator Let $A$ and $B$ be $n\times n$ matrices. Suppose that these matrices have a common eigenvector $\mathbf{x}$. Show that $\det(AB-BA)=0$. Steps. Write down eigenequations of $A$ and $B$ with the eigenvector $\mathbf{x}$. Show that AB-BA is singular. A matrix is […]
  • A Square Root Matrix of a Symmetric Matrix with Non-Negative EigenvaluesA Square Root Matrix of a Symmetric Matrix with Non-Negative Eigenvalues Let $A$ be an $n\times n$ real symmetric matrix whose eigenvalues are all non-negative real numbers. Show that there is an $n \times n$ real matrix $B$ such that $B^2=A$. Hint. Use the fact that a real symmetric matrix is diagonalizable by a real orthogonal matrix. […]
  • Dimension of the Sum of Two SubspacesDimension of the Sum of Two Subspaces Let $U$ and $V$ be finite dimensional subspaces in a vector space over a scalar field $K$. Then prove that \[\dim(U+V) \leq \dim(U)+\dim(V).\]   Definition (The sum of subspaces). Recall that the sum of subspaces $U$ and $V$ is \[U+V=\{\mathbf{x}+\mathbf{y} \mid […]
  • Three Linearly Independent Vectors in $\R^3$ Form a Basis. Three Vectors Spanning $\R^3$ Form a Basis.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 […]
  • $\sqrt[m]{2}$ is an Irrational Number$\sqrt[m]{2}$ is an Irrational Number Prove that $\sqrt[m]{2}$ is an irrational number for any integer $m \geq 2$.   Hint. Use ring theory: Consider the polynomial $f(x)=x^m-2$. Apply Eisenstein's criterion, show that $f(x)$ is irreducible over $\Q$. Proof. Consider the monic polynomial […]
  • Vector Space of 2 by 2 Traceless MatricesVector Space of 2 by 2 Traceless Matrices Let $V$ be the vector space of all $2\times 2$ matrices whose entries are real numbers. Let \[W=\left\{\, A\in V \quad \middle | \quad A=\begin{bmatrix} a & b\\ c& -a \end{bmatrix} \text{ for any } a, b, c\in \R \,\right\}.\] (a) Show that $W$ is a subspace of […]

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