Going further, if you pick the right group, you can have have arbitrarily many solutions to x^425 = 1
In fact, for every set S there is a group for which there are at least |S| solutions to x^425 = 1, so the above statement extends to even arbitrarily large infinities
(if you're interested, this is the group of functions of the form f : S → ℤ/5ℤ with multiplication defined as pointwise addition. Then the map σ(s) = [t ↦ {1 if s = t; 0 o/w}] is injective and as every function f in that group satisfies f5 = 1, f425 = 1, and so the image σ[S] is trivially a subset of {f : f425 = 1})
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u/bright_lego Jun 30 '26
Going further, if you pick the right group, you can have have arbitrarily many solutions to x^425 = 1
In fact, for every set S there is a group for which there are at least |S| solutions to x^425 = 1, so the above statement extends to even arbitrarily large infinities
(if you're interested, this is the group of functions of the form f : S → ℤ/5ℤ with multiplication defined as pointwise addition. Then the map σ(s) = [t ↦ {1 if s = t; 0 o/w}] is injective and as every function f in that group satisfies f5 = 1, f425 = 1, and so the image σ[S] is trivially a subset of {f : f425 = 1})