Math Problem Statement
Let Z60 be the group of integers modulo 60 . For each subgroup below, calculate the index of this subgroup:
the subgroup ⟨17⟩ has index [Z60:⟨17⟩]=
Solution
To calculate the index of the subgroup in , we need to find the number of cosets of in .
The index of a subgroup in is given by the formula:
where is the order of the element in , which is the smallest positive integer such that .
Step-by-Step:
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Find the order of 17 in : The order of 17 is the smallest such that: This implies that is divisible by 60. To find the order, we calculate the greatest common divisor (gcd) of 17 and 60. Since 17 is a prime number and does not divide 60, we can use the fact that the order of an element in is the least number such that , or equivalently, must be a multiple of 60.
The order of in is the least such that:
So, the order of is (60, this number helps
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Math Problem Analysis
Mathematical Concepts
Group Theory
Modular Arithmetic
Subgroups
Index of a Subgroup
Formulas
[Z_n : ⟨a⟩] = n / ord(a)
ord(a) is the smallest k such that a * k ≡ 0 (mod n)
Theorems
Lagrange’s Theorem
Suitable Grade Level
Undergraduate Mathematics
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