Can $\frac{SO(5)}{N}= SU(2) \times SO(2)$ be true?

Asked Dec 07 '16 at 00:01

Active Dec 07 '16 at 00:11

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Inspired by this, consider the unknown short exact sequence,

$$ 1 \to N \to SO(5) \to SU(2) \times SO(2) \to 1 $$

What is the normal subgroup $N$ here so that $SU(2) \times U(1)$ is a quotient group for the total group $SO(50$, and $SO(5)/N= SU(2) \times SO(2)$? Is it an allowed short exact sequence? (Here $SO(2)$ is the same as a $S^1$ circle and a $U(1)$.)

edited Apr 13 '17 at 12:21

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asked Dec 07 '16 at 00:01

wonderich

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$SO(5)$ is a simple Lie group; it has no nontrivial quotients whatsoever. – Qiaochu Yuan Dec 07 '16 at 00:22
Thank you, the expert! How about $O(5)$, $O(4)$ and $SO(4)$? This is practical for elementary particle physics. – wonderich Dec 07 '16 at 00:26
@QiaochuYuan is the same true for $SO(n)$ for odd $n$ in general? – Ben Grossmann Dec 07 '16 at 00:37
@QiaochuYuan, how can we determine whether O(n) or SO(n) are simple or not? I think O(n) has a chance to fulfill this exact sequence??? – wonderich Dec 07 '16 at 00:59

Can $\frac{SO(5)}{N}= SU(2) \times SO(2)$ be true?

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