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Keith Nicholson - Introduction to Abstract Algebra, Good Exercises

2018. 1. 11. 22:12수학 이론/추상대수학

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

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 $$n$$.

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