← MATH 01340 · Modern Algebra I
MATH 01340 · Groups
Cosets / Lagrange's theorem idea
Definition
Left coset aH={ah|h∈H}. Cosets partition G. Lagrange: |H| divides |G| for finite groups; index [G:H]=|G|/|H|.
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aH=bH iff a⁻¹b ∈ H. Cosets need not be subgroups.
Corollaries: order of an element divides |G|; groups of prime order are cyclic.
Normal subgroups (aH=Ha) enable quotient groups — often next chapter.
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1.In S₃, take H=⟨(12)⟩ and list left cosets.