Does o(ab)=lcm(o(a),o(b)) for Group G Elements a & b?

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SUMMARY

The relationship between the order of the product of two group elements and their individual orders is established in group theory. Specifically, for elements a and b in a group G, the equation o(ab) = lcm(o(a), o(b)) holds true if and only if a and b commute. When a and b do not commute, this relationship fails, as demonstrated by counterexamples in the symmetric group S3, where specific elements can yield different orders.

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  • Understanding of group theory concepts, particularly group elements and their orders.
  • Familiarity with the Least Common Multiple (LCM) and its mathematical properties.
  • Knowledge of commutative and non-commutative operations in algebra.
  • Basic understanding of symmetric groups, specifically S3.
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Suppose G is a group, a,b are two elements of G, does o(ab)=lcm(o(a),o(b))?
o(ab) denotes the order of ab, lcm(o(a),o(b)) denotes the Least Common Multiple of o(a) & o(b).
 
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It is true when a and b commute, and can easily be proven in that case.
When a and b do not commute, the statement is false. For a counterexample, you could try to find some elements in S3 (e.g. there are elements a and b such that o(a) = 2, o(b) = 3, o(a, b) = 2).
 

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