Let G be a group. A Weak Cayley Table mapping ϕ : G → G is a bijection such that ϕ(g1g2) is conjugate to ϕ(g1)ϕ(g2) for all g1, g2 in G. The set of all such mappings forms a group W(G) under composition. We study W(G) for the seventeen wallpaper groups G.

Let G be a group. A weak Cayley table isomorphism $\phi$: G $\rightarrow$ G is a bijection satisfying two conditions: (i) $phi$ sends conjugacy classes to conjugacy classes; and (ii) $\phi$(g1)$\phi$(g2) is conjugate to $\phi$(g1g2) for all g1, g2 in G. The set of all such mappings forms a group W(G) under composition. We study W(G) for fifty-six of the two hundred nineteen three-dimensional crystallographic groups G as well as some other groups. These fifty-six groups are related to our previous work on wallpaper groups.

We investigate weak Cayley table isomorphisms, a generalization of group isomorphisms. Suppose G and H are groups. A bijective map phi : G to H is a weak Cayley table isomorphism if it satisfies two conditions:(1) If x is conjugate to y, then phi(x) is conjugate to phi(y); (2) For all x, y in G, phi(xy) is conjugate to phi(x)phi(y).If there exists a weak Cayley table isomorphism between two groups, then we say that the two groups have the same weak Cayley table.This dissertation has two main goals. First, we wish to find sufficient conditions under which two groups have the same weak Cayley table. We specifically study Frobenius groups and groups which satisfy the Camina pair condition. Second, we consider the group of all weak Cayley table isomorphisms between G and itself. We call this group the weak Cayley table group of G and denote it by W(G). Any automorphism of G is an element of W. The inverse map on G is also an element of W. We say that the weak Cayley table group is trivial if it is generated by the set of all automorphisms of G and the inverse map. Stephen Humphries proved that the symmetric groups S_n, the dihedral groups D_{2n} and the free groups F_n (n not equal to 3) all have trivial weak Cayley table groups. We will investigate the weak Cayley table groups of the alternating groups, certain types of Coxeter groups, the projective special linear groups and certain sporadic simple groups.