A monodromy group is the group of transformations obtained by monodromy around loops
based at a fixed point. For an algebraic function
with 
branches at the base point, analytic
continuation around loops avoiding exceptional values
gives a permutation group,
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where 
is the domain of continuation, 
is its fundamental
group, and 
is the symmetric group. The monodromy group is
the image of this group
homomorphism, and relabeling the branches conjugates
it in 
(McMullen 2026, pp. 27-28). For 
with a positive integer 
, it is the cyclic
group generated by a cyclic permutation
of all 
branches.
When 
is the complex plane with all exceptional values
removed, an irreducible polynomial 
over 
gives a transitive group
action on its 
branches. Any one branch can be
continued into any other along a suitable loop (Puiseux
1851, pp. 233-235).
For a system of linear ordinary differential equations,
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for 
, ..., 
, where the 
are analytic functions
on a complex domain 
, analytic continuation
acts on the solution vector space. Choosing a basis represents these transformations by invertible matrices
in the general linear group 
. Their group is the
monodromy group of the system. Changing the solution basis
conjugates all the matrices by the same invertible matrix (Singer 2008, p. 19).
