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What is the smallest $m \mid 60$ so that there is no subgroup $H \leq A_5$ with order $m$ ?

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    indeed there exists one. so m must be 15. thanks guys ! i thought wrong over this problem.2012-10-06

2 Answers 2

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The factors of $60$, in order, are $1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60$.

It's easy to see that $A_5$ has cyclic subgroups orders $1, 2, 3$ and $5$, and $\langle (12)(34),(13)(24)\rangle$ is a subgroup of order $4$.

For $m=6$ or $10$, the possible groups of order $m$ are cyclic and dihedral. $A_5$ has no element of either order, so we need to look for dihedral subgroups. Thus we seek elements of order $3$ and $5$ which are conjugated into their inverses by appropriate elements of order $2$. These are easily found: $\langle (123), (12)(45)\rangle$ has order $6$ and $\langle (12345),(14)(23)\rangle$ has order $10$.

The five copies of $A_4$ each have order $12$.

Every group of order $15$ is cyclic, and (by considering possible cycle types) $A_5$ has no element of order $15$, so $A_5$ has no subgroup of order $15$, and $15$ is the smallest such $m$.

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    is every group with$15$elements isomorph to $\mathbb Z / 15 \mathbb Z$ ?2012-10-08
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First,Using Sylow's theorem,we can rule out $2,3,5$

Now considering $D_{10}$ (Dihedral group),we can rule out $10$

Considering $A_4$ , we can see that $m \geq 12$.

Now the next possible value is $15$ ,please check the following thread Why $A_{5}$ has no subgroup of order $15$?

Not sure it is the best method though,there might be some way to find it without explicitly checking for every value of $m$.

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    $A_5$ is the symmetry group of the pentagondodekahedron and as such contains $C_5$ and $D_5$ (the stabilizer of a face, the stabilizer of two opposite faces).2012-10-06