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 \(n\).