Question

State the following statement are True? 1. If G = <x> is of order 25 , then x^(alpha) generates G , where alpha is a factor of 25

Answer 1

Answer on Question #75453 - Subject – Abstract Algebra

**Given:** $G = \langle x \rangle$ and $o(G) = 25$

**To prove or disprove:** $G = \langle x^{\alpha} \rangle$ where $\alpha$ is a factor of 25.

**Solution:** Consider $G = \langle x \rangle$ and $o(G) = 25$

$\Rightarrow G$ is a cyclic group and generated by $x$.

$\therefore x^{25} = e$ and $o(x) = 25$

$\therefore$ The order of an element in $G$ can be 1, 5 or 25.

Let $x^{\alpha}$ generate the group $G$ and $\alpha$ is a factor of 25, therefore

$o(x^{\alpha}) = 25$ and $25 = \alpha k$, where $k < 25$ is an integer

Q $x^{25} = e$ $\Rightarrow x^{\alpha k} = e$

$\Rightarrow (x^{\alpha})^{k} = e$

$\Rightarrow k = 25$

But $k \neq 25$. Therefore $\alpha$ can't be a factor of 25.

Hence, given statement is false.

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