Filling Gaps of the Symmetric Crosscap Spectrum

Symmetric Crosscap Number of Groups of Order Less Than or Equal to 63

“…So, it is a group that acts on a nonorientable surface. • For the NEC group (0; +; [4]; {(2,2,2)}), with reduced area 1/2, we have that an associated epimorphism θ : Λ → G is given by…”

“…A presentation of this group is given by generators a, b, c and relations a 4 = b 3 = c 3 = 1, ba = ab 2 and ca = ac 2 , given by GAP. For the NEC group (0; +; [3,4,4]; {(-)}), with reduced area 13/12, we have that an associated epimorphism θ : Λ → G is given by…”

“…This presentation is given by GAP, because if we see the presentation in [13] we can observe that there is a misprint in the fourth relation, as the last exponent should be a −2 instead of a 2. For the NEC group (0; +; [4,4]; {(-)}), with reduced area 1/2, we have that an epimorphism θ : Λ → G is given by…”

“…Let us study the element ab. We have that (ab) 4 = ab(ab) 2 ab = ab(ba) 2 ab = ab 2 aba 2 b = a 2 b −2 ba 2 b = a 2 b −1 a 2 b = a 2 b −1 ba 2 = 1; so that ab has order 4. Take the element x 1 , whose image is the generator a −1 , and the element x 2 1 x 2 that has as image the generator b.…”

“…Note that the algebraic structure of the group [60,1], C 5 × DC 3 , that was studied in [11] can be expressed as C 3 C 20 . Hence, this group belongs to the family C 3 C 8k+12 , whose symmetric crosscap numbers fill the numbers of the form 24k + 15; see [4].…”