Cosets and quotients
Objective. Apply Lagrange's theorem and construct quotients.
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- 2. Worked example
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Learn: the key idea
Cosets and quotients partition a group G by a subgroup H into left cosets gH = {gh : h ∈ H}, all of equal size |H|, giving Lagrange's theorem: |H| divides |G|. When H is normal (gHg⁻¹ = H for all g), the cosets themselves form a group G/H under (aH)(bH) = abH, called the quotient group. For G = ℤ and H = 3ℤ, the cosets are 0+3ℤ, 1+3ℤ, 2+3ℤ, and the quotient ℤ/3ℤ has exactly 3 elements, matching |ℤ/3ℤ| as expected from Lagrange's theorem reasoning in the finite case.
Worked example
Using Lagrange's theorem, determine the possible orders of subgroups of a group of order 12.
- 1Lagrange's theorem states that the order of any subgroup must divide the order of the group.
- 2List the positive divisors of 12: 1, 2, 3, 4, 6, 12.
- 3Each divisor is a candidate order, though not all must necessarily be realised by an actual subgroup.
- 4So the possible subgroup orders are exactly these divisors of 12.
Answer: 1, 2, 3, 4, 6, 12
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Start practisingCommon mistake. Students assume the quotient construction works for any subgroup, but aH·bH = abH is only well-defined when H is normal; using a non-normal subgroup produces inconsistent coset products depending on chosen representatives.
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