# Yu-Tsumura-small

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- Group of Order 18 is Solvable Let $G$ be a finite group of order $18$. Show that the group $G$ is solvable. Definition Recall that a group $G$ is said to be solvable if $G$ has a subnormal series \[\{e\}=G_0 \triangleleft G_1 \triangleleft G_2 \triangleleft \cdots \triangleleft G_n=G\] such […]
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- Polynomial $(x-1)(x-2)\cdots (x-n)-1$ is Irreducible Over the Ring of Integers $\Z$ For each positive integer $n$, prove that the polynomial \[(x-1)(x-2)\cdots (x-n)-1\] is irreducible over the ring of integers $\Z$. Proof. Note that the given polynomial has degree $n$. Suppose that the polynomial is reducible over $\Z$ and it decomposes as […]
- Determine Whether Given Subsets in $\R^4$ are Subspaces or Not (a) Let $S$ be the subset of $\R^4$ consisting of vectors $\begin{bmatrix} x \\ y \\ z \\ w \end{bmatrix}$ satisfying \[2x+4y+3z+7w+1=0.\] Determine whether $S$ is a subspace of $\R^4$. If so prove it. If not, explain why it is not a […]
- The Order of $ab$ and $ba$ in a Group are the Same Let $G$ be a finite group. Let $a, b$ be elements of $G$. Prove that the order of $ab$ is equal to the order of $ba$. (Of course do not assume that $G$ is an abelian group.) Proof. Let $n$ and $m$ be the order of $ab$ and $ba$, respectively. That is, \[(ab)^n=e, […]
- Vector Form for the General Solution of a System of Linear Equations Solve the following system of linear equations by transforming its augmented matrix to reduced echelon form (Gauss-Jordan elimination). Find the vector form for the general […]
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