Keith Nicholson - Introduction to Abstract Algebra, Good Exercises

Keith의 책으로 현대대수를 공부하고 있는데, 연습문제 중에 괜찮은 개념을 담고 있거나 난도가 있는 문제들을 따로 정리해 두기로 한다.

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CH 2.4. Cyclic Groups and Order of an element

24.

(a) $h$ is the only element of $\text{order 2}$ in a group $G$. show that $h \in Z(G)$, where $Z(G)$ is the center of group $G$.

(b) $k$ is the only element of $\text{order 3}$ in a group $G$. What can you say about $k$?

35.

(a) Let $a,b$ are elements of a group $G$, and let $m,n$ be $\text{ord}(a)$ and $\text{ord}(b)$, respectively. 

If $ab = ba$, show that $G$ has an element $c$, such that $\text{ord}(c) = \text{lcm}(m,n)$.

(b) Let $G$ be an abelian. And assume that $G$ has an element of maximal order $n$. 

Show that $\forall g \in G, \ g^n = 1$.

37.

Faro shuffle of a deck which contains $2n$ cards can be represented by the permutation $\phi$ : 

$$\phi = \begin{pmatrix} 1 & 2 & 3 & 4 & \cdots & 2n \\ 1 & n+1 & 2 & n+2 & \cdots & 2n \end{pmatrix}$$

Let $m \ge 1$ be a minimum integer such that $\iota = \phi^{m}$, where $\iota$ is the identity permutation. 

Express $m$ by terms about $n$.