Kernel and image of a homomorphism $SL(2,5)\to S_5$

Asked Jan 08 '15 at 13:38

Active Jan 08 '15 at 14:02

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Since $SL(2,5)$ has a subgroup of index $5$, I can use the left coset action to define a homomorphism between $SL(2,5)$ and $S_5$. How can I find the kernel and the image of this homomorphism?

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The kernel would be a normal subgroup of $SL(2,5)$. Other than the trivial ones there's the cyclic group of order $2$, but I'm not sure these are all of them.

edited Jun 12 '20 at 10:38

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asked Jan 08 '15 at 13:38

highlander

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Well, what are the possible kernels (i.e. what are the normal subgroups)? – Tobias Kildetoft Jan 08 '15 at 13:46
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@TobiasKildetoft Other than the trivial ones there's the cyclic group of order 2, but I'm not sure these are all of them. – highlander Jan 08 '15 at 13:57
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$S_5$ has a trivial center. – i. m. soloveichik Jan 08 '15 at 14:08
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related: Elementary isomorphism between $PSL(2,5)$ and $A_5$ etc – Grigory M Jan 08 '15 at 14:09
This is appreciated, but I still can't quite see it. – highlander Jan 11 '15 at 22:25

Kernel and image of a homomorphism $SL(2,5)\to S_5$

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