A Is SU(3) always contains SU(2) groups?

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In the discussion, the user seeks clarification on whether non-trivial SU(3) groups always contain SU(2) subgroups, and similarly, if SU(2) groups contain U(1) subgroups. They note the existence of multiple SU(2) subgroups within SU(3) and compare this to the relationship between O(2) and O(3). The user questions the compatibility of SU(3) with the product group SU(3) x SU(2) x U(1), emphasizing that SU(3) has fewer parameters than the product group. They also express confusion regarding the term "compatible" in this context, indicating a need for further explanation.
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Hi,

I trying to understand. If there is non-trivial SU(3) group, is it always possible to find SU(2) as part of SU(3)?
And same question about SU(2) and U(1).
 
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Sure, there are many ##SU(2)## subgroups of ##SU(3)## just like there are many ##O(2)## subgroups of ##O(3)##.
 
Thanks.

And another question about same.
SU(3) seems as have less number of parameters than SU(3)xSU(2)xU(1).
If there is SU(3) group, is is possible to say it is compatible with SU(3)xSU(2)xU(1) because SU(3) always contains SU(2) subgroups, and SU(2) always contains U(1)?
 
The group, ##G_1\times G_2## is the tensor product of two independent groups. ##G_1## is contained in ##G_1\times G_2## by the projection ##\pi : (g1,g2)\rightarrow g1##. The subgroup inclusion discussed in my previous reply is a subgroup of a different sort. (Mathematics is not my strongest subject) It would help to know what you mean by "compatible?"
 
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