Mathematics Exam > Mathematics Questions > If a, b ∈G, a group of order m, then ord...
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If a, b ∈ G, a group of order m, then order of ab and ba are
- a)same
- b)equal to m
- c)unequal
- d)None of these
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
If a, b ∈G, a group of order m, then order of ab and ba area)same...
If a, b are variables, there are many possible mathematical operations that can be performed with them. Some common operations include addition (a + b), subtraction (a - b), multiplication (a * b), and division (a / b). Additionally, variables can be used in more complex operations such as exponentiation (a^b) or taking the square root (√a). The specific operation that should be performed with a and b depends on the context and what you are trying to achieve.
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If a, b ∈G, a group of order m, then order of ab and ba area)same...
- Order of an Element:
The order of an element g in a group G is the smallest positive integer n such that gⁿ = e, where e is the identity. - Relationship Between ab and ba:
- Observe that ab and ba are conjugate elements. Specifically, ba = a⁻¹(ab)a, so ba is conjugate to ab.
- Key Property: Conjugate elements have the same order.
- Proof Using Powers:
- Suppose (ab)ⁿ = e. Then:
(ba)ⁿ = b(ab)ⁿ⁻¹a = b · e · a = ba (if n = 1). - For general n, use induction or note that (ba)ⁿ = a⁻¹(ab)ⁿa. If (ab)ⁿ = e, then (ba)ⁿ = a⁻¹ea = e.
- Similarly, if (ba)ⁿ = e, then (ab)ⁿ = e.
- Thus, the smallest n satisfying (ab)ⁿ = e equals the smallest n for (ba)ⁿ = e.
- Suppose (ab)ⁿ = e. Then:
- Examples in Non-Abelian Groups:
- In S₃, let a = (1 2 3) and b = (1 2). Then ab = (1 3) (order 2) and ba = (2 3) (order 2).
- Another example: a = (1 2), b = (2 3). Then ab = (1 2 3) (order 3) and ba = (1 3 2) (order 3).
Conclusion:
The order of ab and ba are always the same in any group G.
The order of ab and ba are always the same in any group G.
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