A set of generators 
is a set of group elements such that possibly repeated application of the generators
on themselves and each other is capable of producing all the elements in the group.
Cyclic groups can be generated as powers
of a single generator. Two elements of a dihedral group
that do not have the same sign of ordering are generators for the entire group.
The Cayley graph of a group 
and a subset of elements (excluding the identity
element) is connected iff the subset generates the group.
See also
Cayley Graph, Finitely Generated
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References
Arfken, G. "Generators." ยง4.11 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 261-267, 1985.
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Cite this as:
Weisstein, Eric W. "Group Generators." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GroupGenerators.html