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


(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 \(k\)?


(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 \(\forall g \in G, \ g^n = 1\).


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 \(n\).