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Abstract Algebra

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Order 10 Groups Embedded in $S_{10}$

ABSALG-JW8OBH

Let $G$ be a group of order 10 and let $g \in G$ have order 5. By Cayley's theorem, the action of $G$ on itself by left multiplication gives rise to an embedding $G \hookrightarrow S_{10}$.

What is the cycle type of $g$ under this embedding?

(In the answer choices below, the notation $(a_1,a_2,\ldots,a_n)$ refers to an element whose disjoint cycle decomposition is a product of cycles of length $a_1, a_2, \ldots, a_n$).

A

$(5,5)$

B

$(1,1,1,1,1,5)$

C

$(2,2,2,2,2)$

D

$(10)$

E

$(5,2,1,1,1)$