So I have done problem #13 in section 7.5 of t book Abstract Algebra: An Intorduction, 3rd Edition by Thomas W. Hungerford (ISBN: 978-1-1115696-2-4). multiple times over now, but I still get the order of $\alpha$ is 6. Here is some of my work:
$\alpha = (12)(34)(5678910)$ $\alpha^2 = (579)(6810)$ $\alpha^3 = (12)(34)(58)(69)(710)$ $\alpha^6 = \alpha^3 • \alpha^3 = [(12)(34)(58)(69)(710)] • [(12)(34)(58)(69)(710)] = (1)$
Even by Theorem 7.25 I am getting that the order of $\alpha$ is 6. Theorem 7.25 states:
The order of a permutation $\tau$ in $S_n$ is the least common multiple of the lengths of the disjoint cycles whose product is $\tau$.
Is there something I am doing incorrectly, or is the book wrong?
P.S. I tried to post images of my work and the book question but “I need at least 10 reputation to post images” or so the error message said.
You're right. The decomposition into disjoint cycles yields $$(1\, 2\, 3)(2\, 3 \,4)(5\, 6\, 7)(7 \,8\, 9\, 10)=(1\,2)(3\,4) (5\,6\,7\,8\,9\,10).$$