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Just give a name to your word:

sage: w= Word(x^3*y*x*z^2*x)

Then discover some methd related to periods:

sage: w.per<TAB_KEY>

Then look at the documentation of the periods method:

sage: w.periods?

Then construct your dedicated function, knowing that:

sage: w.periods()
[7]

sage: w = Word('ababab')
sage: w.periods()
[2, 4]

sage: w = Word('abcdabcde')
sage: w.periods()
[]

Just give a name to your word:

sage: w= Word(x^3*y*x*z^2*x)

Then discover some methd related to periods:

sage: w.per<TAB_KEY>

Then look at the documentation of the periods method:

sage: w.periods?

Then construct your dedicated own periodicity function, knowing that:

sage: w.periods()
[7]

sage: w = Word('ababab')
sage: w.periods()
[2, 4]

sage: w = Word('abcdabcde')
sage: w.periods()
[]