Every abelian group of even order with a noncyclic Sylow 2-subgroup is known to be R-sequenceable except possibly when the Sylow 2-subgroup has order 8. We construct an R-sequencing for many groups with elementary abelian Sylow 2-subgroups of order 8 and use this to show that all such groups of order other than 8 also have terraces. This completes the proof of Bailey's Conjecture in the abelian case: all abelian groups other than the noncyclic elementary abelian 2-groups have terraces. For odd orders it is known that abelian groups are Rsequenceable except possibly those with noncyclic Sylow 3-subgroups. We show how the theory of narcissistic terraces can be exploited to find R-sequencings for many such groups, including infinitely many groups with each possible of Sylow 3-subgroup type of exponent at most 3 12 and all groups whose Sylow 3-subgroups are of the form Z ρ 3 × Z ρ 9 × Z σ 27 or Z ρ 3 × Z ρ+1 9 × Z σ 27 .
We generalise an extension theorem for terraces for abelian groups to apply to non-abelian groups with a central subgroup isomorphic to the Klein 4-group $V$. We also give terraces for three of the non-abelian groups of order a multiple of 8 that have a cyclic subgroup of index 2 that may be used in the extension theorem. These results imply the existence of terraces for many groups that were not previously known to be terraced, including 27 non-abelian groups of order 64 and all groups of the form $V^s \times D_{8k}$ for all $s$ and all $k > 1$ where $D_{8k}$ is the dihedral group of order $8k$.
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