What can be a flavour symmetry of the Standard model? 
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*What would be a flavour symmetry in the Standard model and how can it be implemented in the quark and/or lepton sector of the Standard model?

*What is the difference between the quark mixing and a flavour symmetry in the quark sector?

*What are the importance and powers of a flavour symmetry?
 A: The Standard Model (SM) was developed when the SU(3) flavour symmetries were discovered in the accumulation of resonances back in the 1960's and 70's.
It was a clear indication of the existence of a quarks structure for the hadrons and eventually it developed to the SM. In an extension of the SU(2) isotopic spin symmetries, i.e. where the neutron and the proton were considered as one nucleon with isotopic spin 1/2 and -1/2 respectively, it was found that the hadronic systems could be organize in SU(3) representations, and the theory then was called the eightfold way.:


The baryon octet. Particles along the same horizontal line share the same strangeness, s, while those on the same diagonals share the same charge, q.

This is a flavor symmetry, because it is organized according to the quantum numbers of the hadrons in niches of the octet SU(3) representation.

historically, it was the reverse: Quarks were motivated by our understanding of flavour symmetry. Specifically: First it was noticed that groups of particles were related to each other in a way that matched the representation theory of SU(3). From that, it was inferred that there is an approximate symmetry of the universe which is parametrized by the group SU(3). Finally, this helped lead to the discovery of quarks, three of which are interchanged by these SU(3) transformations (the three lightest: up, down, and strange).

You ask

2)What is the difference between the quark mixing and a flavour symmetry in the quark sector

From the article 

in the Standard Model of particle physics, the Cabibbo–Kobayashi–Maskawa matrix, CKM matrix, quark mixing matrix, or KM matrix is a unitary matrix which contains information on the strength of flavour-changing weak decays. Technically, it specifies the mismatch of quantum states of quarks when they propagate freely and when they take part in the weak interaction.

So it is about the weak interactions of particles.

3)What are the importance and powers of a flavour symmetry?

A flavor symmetry is a relationship between observerd hadrons. In the unifications of weak and electromagnetic interactions with the HIggs mechanism,  the weak SU(3) multiplets can be treated as one "multiplet" analogous to the "one nucleon" when only SU(2) was known for hadrons. 
The quark mixing matrix is necessary in comparing calculations for weak decays with data, because experimentally it was found that the mass eigenstates and the eigenstates describing weak decays were different. It is simpler to consider the Cabibbo angle ( a  smaller space subset )
The Cabibbo angle represents the rotation of the mass eigenstate vector space formed by the mass eigenstates | d ⟩ ,   | s ⟩  into the weak eigenstate vector space formed by the weak eigenstates | d ′ ⟩ ,   | s ′ ⟩ . θC = 13.02°.
Both the flavor symmetry and the mixing matrix are part of the formulation of the  very successful  Standard Model which models the interactions of elementary particles . The recent discovery of the Higgs at LHC is part of the power of the SM.
