Let \(G\) be a finite group of order \(o(G)\) and \(S\) be a generating set for \(G\). Let \(t_ 2(S)\), \(t_ 3(S)\) and \(t_ h(S)\) be the number of generators in \(S\) of order 2, 3 and of order larger than 3 respectively; \(| S|:= t_ 2(S)+ t_ 3(S)+ t_ h(S)\). \(\psi(G)\) is defined as the minimum of \(| S|\), where \(S\) is a generating set for \(G\). Examples:
a) If \(H_ n\) is the dicyclic group of order \(4n\), then \(\psi(H_ n)= 18\), for \(n\neq 3\) and \(\psi(H_ 3)= 17\). b) If \(G= (\mathbb{Z}_ 2)^ a\times H_ n\) with \(a\geq 1\) and \(n\) even, then \(\psi(G)= 3a+ 18\). c) Let \(M_ t= (3,3| 3,t)\) of order \(3t\) with \(t\geq 3\); then \(\psi(M_ t)= 16\). d) For the Abelian group \(A= \mathbb{Z}_{m_ 1}\times\cdots\times \mathbb{Z}_{m_ c}\) s.t. \(m_ i\) divides \(m_{i+ 1}\) for \(i= 1,\dots, c- 1\), \(m_ 1> 1\), \(m= m_ 1=\cdots+ m_ a\) with \(a< c\) and \(m_{a+ 1}\neq m_ a\) the function \(\psi\) has the value \(9c- 6a\) if \(m= 2\), \(9c- a\) if \(m\geq 3\) and \(9c\) if \(m\geq 4\). e) If \(n\) is even and \(D_ n\) is the dihedral group, the \(\psi((\mathbb{Z}_ 2)^ a\times D_ n)= 3a+ 6\) for all \(a\geq 1\).
The real genus \(\rho(G)\) is the minimal algebraic genus of any compact bounded Klein surface on which \(G\) is acting. The main result of this paper is the following lower bound for \(\rho(G)\): \[ \rho(G)\geq 1+ o(G) [\psi(G)- 12]/12. \] This lower bound is sharp, because it is attained for some groups. The author applies the above formula to determine the real genus of several infinite families of groups such as:
i) \(\rho((\mathbb{Z}_ 1)^ a\times H_ n)= \begin{cases} 1+ 2^ a(a+ 2)n & \text{if }n\text{ is even },\;a= 1\text{ or }a= 2,\\ 1+ 2^ a(a+ 3)n &\text{ if }n\text{ is even },\;a= 3.\end{cases}\)
ii) \(\rho(M_ t)= 1+ t^ 2\) if \(t\geq 3\).
iii) \(\rho((\mathbb{Z}_ 3)^ 3)= 1+ 3^{c- 1}(2c- 3)\) for \(c\geq 1\).
iv) \(\rho((\mathbb{Z}_ 4)^ c)= 1+ 4^{c- 1} (3c- 4)\) for \(c\geq 1\).
v) If the Abelian group \(A\) has the canonical form \((\mathbb{Z}_ 2)^ a\times \mathbb{Z}_{2m_ 1}\times\cdots\times \mathbb{Z}_{2m_ b}\), where \(a> b\geq 0\) and \(m_ 1> 1\), then \[ \rho(A)= 1+ \sigma(A) (3b+ a- 3)/4. \] Particularly: \(\rho((\mathbb{Z}_ 2)^ a)= 1+ 2^{a- 2}(a- 3)\).
vi) \(\rho((\mathbb{Z}_ 2)^ a\times D_ n)= 1+ 2^{a-1} (a- 1)n\), where \(a\geq 1\) and \(n\) is even.
Finally, some unsolved problems about the real genus of certain groups are indicated.