Step 1: Problem Overview:
This question tests your knowledge of a key theorem in finite group theory. The task is to identify the theorem based on its description, which links a prime factor of a group's order to the presence of an element with that prime order.
Step 2: Theorem Analysis:
Let's examine the provided theorems:
Lagrange's Theorem: This theorem states that the order of any subgroup \(H\) of a finite group \(G\) divides the order of \(G\). A consequence of this is that the order of any element in \(G\) divides the order of \(G\). However, it doesn't guarantee the existence of an element with a specific order. The Klein four-group, for instance, has order 4, but no element of order 4.
Sylow's Theorems: These theorems provide information about subgroups of order \(p^k\), where \(p^k\) is the highest power of a prime \(p\) that divides the group's order. They are related to, but more general than, Lagrange's Theorem. Sylow's First Theorem implies Cauchy's Theorem.
Euler's Theorem: This is a number theory theorem, not a group theory theorem. It states that if \(n\) and \(a\) are coprime positive integers, then \(a^{\phi(n)} \equiv 1 \pmod{n}\), where \( \phi \) is Euler's totient function.
Cauchy's Theorem: This theorem asserts that if \(G\) is a finite group and \(p\) is a prime number that divides the order of \(G\), then \(G\) contains an element of order \(p\). This aligns directly with the statement in the problem.
Step 3: Solution:
The correct answer is Cauchy's Theorem.