Question

If p is a prime number and O(G) denotes the order of a group G and p divides O(G), then group G has an element of order p. Then, this is a statement of

• A. Lagrange's Theorem
• B. Sylow's Theorem
• C. Euler's Theorem
• D. Cauchy's Theorem

Hint:

It's essential to distinguish between these key theorems:
\textbf{Lagrange:} Order of subgroup divides order of group. (Goes "down": element \(\to\) group)
\textbf{Cauchy:} Prime divisor \(p\) of group order implies element of order \(p\) exists. (Goes "up": group \(\to\) element)
\textbf{Sylow:} Prime power divisor \(p^k\) of group order implies subgroup of order \(p^k\) exists.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This question asks to identify a fundamental theorem in finite group theory based on its statement. The statement connects a prime divisor of the order of a group to the existence of an element of that prime order.

Step 2: Detailed Explanation:
Let's review the theorems listed:
Lagrange's Theorem: States that for any finite group G, the order of every subgroup H of G divides the order of G. A corollary is that the order of any element of G divides the order of G. It does not guarantee the existence of an element of a certain order. For example, the Klein four-group has order 4, but no element of order 4.
Sylow's Theorems: A set of theorems that guarantee the existence and properties of subgroups of order \(p^k\) where \(p^k\) is the highest power of a prime \(p\) dividing the group's order. They are a partial converse to Lagrange's Theorem. While Sylow's First Theorem implies Cauchy's Theorem, the statement itself is a more general one.
Euler's Theorem: A theorem from number theory, not group theory. 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: States 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 is the exact statement given in the question.
Step 3: Final Answer:
The statement is Cauchy's Theorem.