Tell me is this abuse or discipline?

My daughters friends mom and dad i think are abusive but others say its discipline. This friend is a 12 year old girl who we must admit is a bit sassy. She really rebellious but that does not surprise me as many children are. So she was at her house apparently and she likes to dru her fingers on the wall and this annoyed her 6 year old sister so the sister told on her. Her mom came up and sent them both to their bedrooms and the 12 year old went up the stairs and said "This is why i hate you allison" So her mom dashes up the stairs and into the big sisters room and says "This is why we hate you chelsea" and hits her in the arm 3 times to the point were it is black and blue. They sit in a room and their mom calls them and the oldest is clutching her arm(The one her mom hit) and is crying. Her mom just yells at her for crying and and makes fun of her for crying because her and a friend had a rubber band snapping war and she had welts from the rubber band but, the mom hit her in the welts so it hurt obviously. The oldest just sits and cries and looks at her mom just angry. Seriously what is your opinion on this. Oh by the way these are not the girls real names i switched them for their privacy. My daughter tells me all of this because when stuff like this happens they can always text each other because they trust each other so this is true unless the daughter is lying.