math 92--93的第47题怎么做的？

iltjyUID: 2110484

if the finite group G contains a subgroup of order seven but no element(other than the identity) is its own inverse ,then the order of G could be

答案是35

bvbUID: 194706

first eliminate choice A and D for 7 is not a factor of 27 or 37. (Lagrange Theorem)
Then eliminate B and E for they are both group of even order. A group of even order always has an element whose inverse is itself. Take Z-2n, group of integers modulo 2n, as an example:
n+n=0, which implies n^2=identity, then by n^(n^-1)=identity, n=n^-1.

[ Last edited by bvb on 2005-11-9 at 11:59 ]

wjk302UID: 183696

首先，G的阶要能被7整除
另外，由于no element(other than the identity) is its own inverse，即G不含二阶元，所以G的阶不能被2整除（利用sylow定理）

[ Last edited by wjk302 on 2005-11-9 at 12:26 ]

iltjyUID: 2110484

多谢，虽然知道定理却总想不到用，抽象代数还真成问题。