In a Group, every element has an inverse element. If you drop this property, you get a Monoid.[0] Functions under function composition form a Monoid (or Monoidal Category). Brian Beckman did a nice intro video to this at MS Channel 9 a few years back.[1]

[0] https://en.wikipedia.org/wiki/Monoid

[1] "Brian Beckman: Don't fear the Monad" https://www.youtube.com/watch?v=ZhuHCtR3xq8