Question

(Cauchy’s Theorem for Abelian groups) If is a finite abelian group and is a prime that divides , prove that contains an element of order .[Hint: Use the Fundamental Theorem to show that has a cyclic subgroup of order ; use Theorem 7.9 to find an element of order .]

Short answer

It is proved that, $\mathrm{G}$contains an element of order $\mathrm{p}$.

Step-by-step solution

Fundamental Theorem and Theorem 7.2,7.9

Every finite abelian group $\mathrm{G}$ is the direct sum of cyclic groups, each of prime power orders.

Theorem 7.2 states that the non-zero elements of a field form an abelian group under multiplication.

Theorem 7.9 states that be a group and an element of finite order n then,

1. role="math" localid="1657297632189" ${\mathrm{a}}^{\mathrm{k}}=\mathrm{e}$if and only if role="math" localid="1657297673488" $\mathrm{n}\overline{)\mathrm{k}}$.

2. ${\mathrm{a}}^{\mathrm{i}}={\mathrm{a}}^{\mathrm{j}}$if and only if $\mathrm{i}\equiv \mathrm{j}(\mathrm{mod}\mathrm{n})$.

3.If $\mathrm{n}=\mathrm{td}$, with $d\ge 1$ then $a^t$ has order $\mathrm{d}$.

  contains element of order  

Let $\mathrm{G}$be a finite abelian group and $\mathrm{p}$be a prime that divides $\mathrm{G}$that is $\mathrm{p}\overline{)\left|\mathrm{G}\right|}$, .

By Fundamental Theorem of abelian group, there exists an element $\mathrm{k}\in \mathrm{Z}$and a subgroup $\mathrm{H}$of $\mathrm{G}$such that, $\mathrm{G}\cong {\mathrm{Z}}_{{\mathrm{p}}^{\mathrm{k}}}\oplus \mathrm{H}$ .

Thus, for $\mathrm{a}\in \mathrm{G},\left|\mathrm{a}\right|={\mathrm{p}}^{\mathrm{k}}$. Since $\left|\mathrm{a}\right|\in \mathrm{G}$ implies $\overline{)\mathrm{p}}{\mathrm{p}}^{\mathrm{k}}$ then, the order of a can be written as$\left|{\mathrm{a}}^{{\mathrm{p}}^{\mathrm{k}-1}}\right|=\mathrm{p}$

Here, the element ${\mathrm{a}}^{{\mathrm{p}}^{\mathrm{k}-1}}\in \mathrm{G}$ has order $\mathrm{p}$.

Therefore, if $\mathrm{G}$is a finite abelian group and $\mathrm{p}$ is a prime that divides$\left|\mathrm{G}\right|$ , then $\mathrm{G}$ contains an element of order $\mathrm{p}$.