[Math] How to find cyclic subgroup of group of permutations


I'm taking a introductory class of Abstract Algebra, and is having hard time on applying the definition of a cyclic group onto a group of permutations. Because so far I've only learned how to find cyclic subgroups for groups of multiplication and addition, for which I find either the values of the generator to nth power or n multiply to the generator, such that n belongs to Z. So how is such concept of "n copies of the generator"(the phrase from my teacher) applied or adapted to a group of permutations?

Best Answer

Composition is the group operation. So $$\sigma=(12)(3)$$ in $S_3$ has order 2, e.g.

In general the order of $\sigma$ is the minimum $n$ such that $$ \underbrace{\sigma(\sigma(\cdots \sigma(}_{n~times}\cdot)\cdots))=e $$ where $e$ is the identity permutation. To make it clearer, write the permutation in list notation. So $$\sigma=(213),\sigma^2=(123)=e$$ and the cyclic subgroup is $\{\sigma,e\}.$

Why dont you compute the cyclic group generated by $(134)(25)$ in $S_5$ as an exercise. Hint: It has order 10.

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