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Let $A_{5}$ be the alternating subgroup of the symmetric group $S_{5}$. Prove that $A_{5}$ is generated by the two elements $\{a=(123),b=(12345)\}$, or equivalently can we write the element $(234)$ as a composition of the two elements $a$ and $b$.

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This is a virtual duplicate of another recent question. What have you tried? – Mark Bennet Apr 16 '12 at 18:46
I would also be interested in how you know that the two alternatives are equivalent. – Mark Bennet Apr 16 '12 at 19:32
The group is usually called "Alternating group", not "alternative." – Arturo Magidin Apr 16 '12 at 20:12

For the second question, note that $(123)^2(12345)=(145)$, and try repeating the process.

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yes there is a theorem says that $A_{n}=<(123),(234),(345),......((n-2)(n-1)n)> $ from this theorem we can say that $A_{4}=<x=(123),y=(234),z=(345)>$ \\ it is clear $x=a$, and $z=a^{-1}b$, so one just need to know how y could be written in terms of a,b. – kiranovalobas Apr 20 '12 at 20:20

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