Let be a finite group and let and be distinct elements of order 2 in that generate . Prove that , where .
Since is finite, . By a previous exercise, and satisfy the relations of the usual presentation for ; namely, . Moreover, is generated by and ; we know is generated by and and we have . From our discussion about , then, every element of can be written uniquely as for some and . We can thus define a mapping by . This mapping is well defined since every element of has a unique representation as for some and . Moreover, is a homomorphism since . is also injective, due to the uniqueness of representations of elements in and in terms of and adn and , respectively, and is surjective similarly. Thus .