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Let C be a group with |C| = 44. Prove that C must contain an element of order 2.

Let C be a group with |C| = 44. Prove that C must contain an element of order 2.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Problem 3: Group Theory** Let \( C \) be a group with \( |C| = 44 \). Prove that \( C \) must contain an element of order 2. **Explanation:** In this problem, you are asked to demonstrate that a group \( C \) with 44 elements necessarily includes an element whose order is 2. To solve this, apply **Cauchy’s Theorem** which states that if a prime number \( p \) divides the order of a group, then the group contains an element of order \( p \). Since the number 44 can be factored into prime numbers as \( 44 = 2 \times 2 \times 11 \), the prime number 2 divides 44. Therefore, by Cauchy’s Theorem, the group \( C \) must contain an element of order 2.
Transcribed Image Text:**Problem 3: Group Theory** Let \( C \) be a group with \( |C| = 44 \). Prove that \( C \) must contain an element of order 2. **Explanation:** In this problem, you are asked to demonstrate that a group \( C \) with 44 elements necessarily includes an element whose order is 2. To solve this, apply **Cauchy’s Theorem** which states that if a prime number \( p \) divides the order of a group, then the group contains an element of order \( p \). Since the number 44 can be factored into prime numbers as \( 44 = 2 \times 2 \times 11 \), the prime number 2 divides 44. Therefore, by Cauchy’s Theorem, the group \( C \) must contain an element of order 2.
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