Your friend would be best served by trying to obtain information directly. She can also review the California tenant's guide.

Quote Quoting prince7122002
Now, the roommate is asking for her part of next month's rent because my friend did not give her 30 days of notice. My friend said that since she only gave 20 days of notice, she offered to pay for the 10 days she did not give her of notice...Who is wrong here?
From what you have written, it isn't at all clear that they aren't both offering the same thing. One says "pay part of next month's rent", the other offers ten days, which is part of the next month's rent.

Quote Quoting prince7122002
NI mean, I can see where the roommate is coming from, just because my friend's stuff was still in the apartment until June 4, and I believe rent was already due. And since her stuff was still there, I said she should pay for the 30 days.
Leaving items in a rental unit at the conclusion of a tenancy does not necessarily extend the tenancy. The California Department of Consumer Affairs provides some information on a landlord's remedies when a tenant leaves property behind.

Quote Quoting prince7122002
However, I can see my friend's point, which is that since she only gave 20 days of notice, she will pro-rate the 10 days.
Was the twenty day notice ever accepted as sufficient?

Quote Quoting prince7122002
FYI-the landlord gave my friend permission to take her name off the contract.
If both roommates were on a written lease with a landlord, rent is not owed to the other roomate; it would be owed to the landlord. Further, the landlord would not give somebody "permission to take their name off of the contract" - the landlord would release somebody from the lease.

Quote Quoting prince7122002
Now, the roommate talked with the owner and told the owner the situation, and the owner said that my friend should not get her security deposit back, so it will just be for next month's rent, and then the roommate said that my friend will have to pay the remaining balance. Is that fair? I mean, is that legal?
You can review the handbook linked above in relation to the return of a security deposit.