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Abelian Abelian group arrow associated axis binary operation Cayley diagram closed paths commutative concept congruence motions cosets with respect counter-clockwise cycle defining relations deformed denote dihedral group direct product directed segments distinct elements elements of G equilateral triangle example Exercise factor group finite group flip followed four-group geometric group axioms group element group G group isomorphic group of congruence group of order group theory homomorphic mapping icosahedron implies infinite cyclic group initial point integers inverse labeled Lagrange's theorem left cosets manifold mathematics modulo multiplication table non-commutative normal subgroup odd integers one-one pattern period permutation group plane proper subgroups prove quaternion group reader real numbers relation r3 represent representation right cosets satisfy set of defining set of relations shown in Figure subgroup H subgroup of G subgroup of order subset Suppose symbols symmetric group tetrahedron tion unit element verify vertices