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so we are going to show that every finite group is isomorphic to some quotient of a free group to do that lets first go over what a free group is usually when we look at groups we want to think about the relations between elements for example we might say in the dihedral group that r times s is equal to s times r inverse or if were looking at the symmetric group we might say that one two times two three equals one two three but we can also consider another type of group which is a group where there are no relations between elements at all in other words we dont have any of these kinds of equalities between elements every element is distinct and also every product is distinct the only difference is when we look at a group we also have to satisfy the group axioms so in the case of a free group we always need to have for example g times g inverse equals the identity that never changes because we need it to be a group but if were looking at the free group on a set of elements so suppo
