# Reading symmetry out of Lagrangian

Consider the Lagrangian:

$$\mathcal L=\mathcal L_{kin}+\frac{1}{2}m_a^2(\phi^2_1+\phi^2_2+\phi^2_3)+\frac{1}{2}m_b^2(\phi^2_4+\phi^2_5)+\lambda_a(\phi^2_1+\phi^2_2+\phi^2_3)^2+\lambda_b(\phi^2_4+\phi^2_5)^2$$

When $$\phi_1,\phi_2,\phi_3,\phi_4,\phi_5$$ are real scalar fields.

I have 1 question:

(1) What are the symmetries of the lagrangian?

I try to solve this question but I'm not quiet sure what the symmetries are? In the one hand if we define $$\phi_a=(\phi_1,\phi_2,\phi_3)$$ and $$\phi_b=(\phi_4,\phi_5)$$, we can say that each of them can rotate in their on phase i.e there is $$SO(3)$$ symmetry for $$\phi_a$$ and $$SO(2)$$ symmetry for $$\phi_b$$ so our lagrangian full symmetry is $$SO(3)\times SO(2)$$.

On the other hand, we may say that this is not the most general symmetry, and there is also reflection for each vector (this fact that i`m not sure about) so for $$\phi_a$$ we have $$O(3)$$ symmetry (rotation + reflections) and for $$\phi_b$$ the symmetry is $$O(2)$$, ending up with total symmetry of $$O(3) \times O(2)$$