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As @dan_fulea says, it depends on the class of the group:

sage: search_def('is_cyclic')
groups/abelian_gps/abelian_group.py:1037:    def is_cyclic(self):
groups/additive_abelian/additive_abelian_group.py:372:    def is_cyclic(self):
groups/perm_gps/permgroup.py:3482:    def is_cyclic(self):
plot/arc.py:221:            def is_cyclic_ordered(x1, x2, x3):
rings/finite_rings/integer_mod_ring.py:818:    def multiplicative_group_is_cyclic(self):

The command search_def searches for the given string as part of the definition of a function or method, so you can see that is_cyclic is defined for abelian groups, additive abelian groups, and permutation groups.