i need some upper estimate of number of groups of order 24. i know there are 15 of them, but i need upper estimate with explanation, it means why there cannot be more
This problem is a triviality using Burnside's Lemma. Are you working out of Herstein, Cohn, Jacobson or Dummit-Foote? The Herstein and Jacobson do not treat this topic, but in the Cohn you can find it in the section on Group Actions. Same with Dummit-Foote. I do not immediately see a way to get an upper bound at precisely 15 without the use of Burnside's Lemma, however. For that it will be best to have a working knowledge of solvable groups.
Edit: if this class has an emphasis on combinatorics, a Polya-Redfield equation could probably count it. But that would be more difficult than using the Burnside, and would still require knowledge of group actions (but not solvability criteria). Your choice!