If my tenant gets a cat when the lease says no pets, and then moves out after 7 months of a 1 year lease, can I keep a portion of the security deposit since they violated the lease and brought in a cat?

You can't retain a portion of the security deposit in connection with the cat unless the cat caused damage to the property.

If the tenant broke the lease and moved out early then you have to try to re-rent the property. The tenant is responsible for the rent until a new tenant moves in or the end of the lease, whichever occurs first. You can retain the security deposit to be applied first to damages, then to rent. But you have to send the tenant a written statement of how the security deposit was used within 21 days after they move out.

I suggest that you check out http://www.dca.ca.gov/publications/landlordbook/catenant.pdf so you will know your rights and responsibilities as a landlord.

You could have terminated her tenancy as soon as you found out about the violation. By not taking action you allowed the violation.

As noted, you can only use the deposit to repair damages (including cleaning cat smell) or to cover unpaid rental. Since she left early, she owe until the end of the lease or until you replace her.