UTM Groups and Symmetry 习题求助

hbghlyj

因为上次问的3道习题都解决了,所以另开新帖.

11.9. Let $G$ be a finite abelian group and let $m$ be the least common multiple of the orders of its elements. Prove that $G$ contains an element of order $m$.
11.10. Supply a finite non-abelian group for which the conclusion of the previous exercise fails.
11.11. If $H$ is a subgroup of a finite group $G$, and if $|G|=m|H|$, adapt the proof of Lagrange's theorem to show that $g^{m!}∈H$ for all $g∈G$.
13.5. Let $G$ be a group of order $4n+2$. Use Cauchy's theorem, Cayley's theorem, and Exercise 6.6 to show that $G$ contains a subgroup of order $2n+1$.

hbghlyj

11.9. 对任意$a,b∈G$有$o(ab)=\operatorname{lcm}(o(a),o(b))$,所以$G$的所有元素之积的阶是$G$的所有元素的阶的最小公倍数.(又见这帖)
11.10. $G=S_3,o((12))=2,o((123))=3$,但是$G$中没有元素阶数为6.
11.11. Consider the set $X$ of all cosets of $H$. Then $G$ acts by multiplication on $X$ and so there is a homomorphism $G \to Sym(X)$. By Lagrange's theorem, $g^{m!}$ acts as the identity. In particular $g^{m!}H =H$ and so $g^{m!}\in H$. (这帖和这帖)
13.5.

Czhang271828

13.5. 考虑左正则作用 $g.:h\mapsto gh$, 则 $g.\in S_{4n+2}$. 其中 $g.$ 将 $(g_1,g_2,\ldots, g_{4n+2})$ 置换为 $(gg_1,gg_2,\ldots, gg_{4n+2})$. 记符号同态 $\phi:G.\to \{\pm 1\},g.\mapsto \mathrm{sgn}(g.)$.

由 Cauchy 定理知存在 $g_0\neq e$ 使得 $g_0^2=e$. 显然置换 $g_0.$ 为 $2n+1$ 个不交对换之积, 从而 $\phi(g_0.)=-1$.

故 $\ker\phi $ 为 $G.$ 的 $2n-1$ 阶子群.

注: 元素 $g$ 作为左正则作用时在右下角加点, 即 $g.$ 表示 $G$ 上的置换函数.