Let X=(V,E) be a finite simple connected graph with n vertices and m
edges. A configuration is an assignment of one of two colors, black or white,
to each edge of X. A move applied to a configuration is to select a black
edge ϵ∈E and change the colors of all adjacent edges of ϵ.
Given an initial configuration and a final configuration, try to find a
sequence of moves that transforms the initial configuration into the final
configuration. This is the edge-flipping puzzle on X, and it corresponds to a
group action. This group is called the edge-flipping group WE(X) of
X. This paper shows that if X has at least three vertices,
WE(X) is isomorphic to a semidirect product of
(Z/2Z)k and the symmetric group Sn of degree n, where
k=(n−1)(m−n+1) if n is odd, k=(n−2)(m−n+1) if n is even, and
Z is the additive group of integers.Comment: 19 page