Let ππ be the group of integers {0, 1, 2, β¦ , π β 1} with addition moduloβ¦
2024
Let ππ be the group of integers {0, 1, 2, β¦ , π β 1} with addition modulo π as the group operation. The number of elements in the group π2 Γ π3 Γ π4 that are their own inverses is ________
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Correct answer: 4
Key idea: an element is its own inverse exactly when doubling it gives the identity (2x = 0) in each component.
In Z2, 2x = 0 mod 2 holds for both 0 and 1, so there are 2 such elements.
In Z3 (odd modulus), 2x = 0 mod 3 implies x = 0, so there is 1 such element.
In Z4, 2x = 0 mod 4 gives x = 0 or x = 2, so there are 2 such elements.
Multiply the counts for each component: 2 Γ 1 Γ 2 = 4. The elements are (0,0,0), (1,0,0), (0,0,2), (1,0,2).
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