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> ... Does distributivity of inversion ~ over operation ⋆ follow from the other Abelian group axioms / properties? If so, how? It does. For all x and y: (1)
by FBT 1y ago
> ... Does distributivity of inversion ~ over operation ⋆ follow from the other Abelian group axioms / properties? If so, how?
It does. For all x and y:
(1) ~x ⋆ x = 0 (definition of the inverse)
(2) ~y ⋆ y = 0 (definition of the inverse)
(3) (~x ⋆ x) ⋆ (~y ⋆ y) = 0 ⋆ 0 = 0 (from (1) and (2))
(4) (~x ⋆ ~y) ⋆ (x ⋆ y) = 0 (via associativity and commutativity)
In (4) we see that (~x ⋆ ~y) is the inverse of (x ⋆ y). That is to say, ~(x ⋆ y) = (~x ⋆ ~y). QED.
- OjotCewIo 1y agoAwesome, thanks! :)
- stephencanon 1y agoRight. Another way to see this is that for a general (possibly non-Abelian) group, the inverse of xy is y⁻¹x⁻¹ (because xyy⁻¹x⁻¹ = x1x⁻¹ = xx⁻¹ = 1 [using "1" for the identity here, as is typical for general groups], or more colloquially, "the inverse operation of putting on your socks and shoes is taking off your shoes and socks"). For an Abelian group, y⁻¹x⁻¹ = x⁻¹y⁻¹, and we're done.