The statement "Every group of order $pqr$ is cyclic" is not true in general. Whether a group of order $pqr$ is cyclic depends on the relationships between $p$, $q$, and $r$, where $p$, $q$, and $r$ are distinct prime numbers.

Key Facts:

1. Cyclic Groups: A group of order $n$ is cyclic if there exists a single element (a generator) such that every other element can be expressed as a power of this generator.
2. Order of Elements and Lagrange's Theorem: The order of any subgroup must divide the order of the group.
3. Sylow's Theorems: The distribution of subgroups of order $p$, $q$, and $r$ heavily influences whether the group is cyclic or not.

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Detailed Analysis:

1. If $pqr$ satisfies certain divisibility conditions, the group can be cyclic:

• If $p < q < r$ are distinct primes, the group can be cyclic if, for each prime $p, q, r$, the number of Sylow subgroups satisfies: $n_p = 1, \; n_q = 1, \; n_r = 1$ This ensures a unique Sylow subgroup for each prime divisor, making the group cyclic.

2. Counterexamples:

• If $p, q, r$ do not satisfy the above uniqueness conditions, there might be non-cyclic groups. For example, groups of order $30 = 2 \cdot 3 \cdot 5$ can be cyclic, but not all such groups are guaranteed to be cyclic.

3. Additional Conditions:

• If $\gcd(p-1, q)$, $\gcd(p-1, r)$, and $\gcd(q-1, r)$ allow proper subgroup structures, non-cyclic groups may exist.

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Conclusion:

Not all groups of order $pqr$ are cyclic. It depends on the interplay of the primes and the structure of the Sylow subgroups.

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Would you like to see specific examples, theorems, or further proof? Here are related questions to expand understanding:

1. How do Sylow theorems classify groups of composite order?
2. What are some concrete examples of groups of order $30$ and their structures?
3. Why does the existence of unique Sylow subgroups guarantee a cyclic group?
4. What are the properties of the direct product of cyclic groups?
5. Are there general rules for when groups of order $p^n \cdot q^m$ are cyclic?

Tip: Always use Sylow theorems to study the subgroup structure when analyzing groups of composite order.