What is the order of the largest cyclic subgroup of Z6 ⊕Z10 ⊕Z15 ? What is the order of the larg

step 1 In this question we have three distinct complex numbers, Z1, Z2 and Z3. These numbers are in ari

What is the order of the largest cyclic subgroup of Z6 ⊕Z10...

What is the order of the largest cyclic subgroup of $Z_{6} \oplus Z_{10} \oplus Z_{15}$ ? What is the order of the largest cyclic subgroup of $Z_{n_{1}} \oplus Z_{n_{2}} \oplus \cdots$ $\oplus Z_{n_{k}} ?$

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Key Concepts

Direct Sum of Finite Cyclic Groups

The direct sum (or direct product) of cyclic groups is a construction where elements are tuples, each taken from a cyclic group, and the group operation is performed component-wise. This concept is fundamental in understanding the structure of finite abelian groups, as they can often be decomposed into a direct sum of cyclic subgroups.

Order of an Element in a Direct Sum

In a direct sum of cyclic groups, the order of an element is determined by the orders of its individual components. Specifically, the order of an element formed by a tuple is the least common multiple (LCM) of the orders of the elements in each coordinate. This method of finding orders is crucial when analyzing the subgroup structure of the group.

Least Common Multiple (LCM) in Group Theory

The least common multiple (LCM) is a key arithmetic concept that finds application in group theory, particularly in determining the order of an element in a direct sum of cyclic groups. By taking the LCM of the orders of the constituent elements, one can ascertain the maximum order possible for elements in the group, which in turn defines the order of the largest cyclic subgroup.

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