<p>A property ℘ is defined to be a nonempty isomorphism-closed subclass of the class all finite simple graphs. A set <em>S</em> vertices graph <em>G</em> said ℘-set if <em>G</em>[<em>S</em>]∈ ℘. The maximum and minimum cardinalities are denoted by <em>M</em><sub>℘</sub>(<em>G</em>) <em>m</em>&l...

When N is a normal subgroup of G, can we reconstruct G from N and G/N? In general, no. For instance, the groups Z/(p2) and Z/(p) × Z/(p) (for prime p) are nonisomorphic, but each has a cyclic subgroup of order p and the quotient by it also has order p. As another example, the nonisomorphic groups Z/(2p) and Dp (for odd prime p) have a normal subgroup that is cyclic of order p, whose quotient is...