Math Problem Statement

if each element, except the identity of a group be of order 2, then prove that the group is abelian.

Solution

We are asked to prove that if each element of a group GG, except the identity element, has order 2, then the group is abelian.

Step-by-Step Proof:

  1. Given Conditions:

    • Let GG be a group, and ee represent the identity element of GG.
    • For every xGx \in G where xex \neq e, we have x2=ex^2 = e. This means that each non-identity element of GG is its own inverse.
  2. Show that GG is Abelian: To prove that GG is abelian, we need to show that for any two elements a,bGa, b \in G, we have ab=baab = ba.

  3. Consider the product abab: Since a2=ea^2 = e and b2=eb^2 = e, let’s compute the product (ab)2(ab)^2: (ab)2=abab=a(ba)b(ab)^2 = ab \cdot ab = a \cdot (b \cdot a) \cdot b Since (ab)2=e(ab)^2 = e by the condition that all non-identity elements have order 2, we now have: a(ba)b=ea \cdot (b \cdot a) \cdot b = e

  4. Manipulating the equation: Now, multiply both sides of the equation by aa on the left and bb on the right: a(ba)b=ea(ba)ba1b1=ea \cdot (b \cdot a) \cdot b = e \quad \Rightarrow \quad a \cdot (b \cdot a) \cdot b \cdot a^{-1} \cdot b^{-1} = e Simplifying this, since a2=ea^2 = e and b2=eb^2 = e, we get: ba=abb \cdot a = a \cdot b Thus, ab=baab = ba, which shows that GG is abelian.

Conclusion:

Therefore, the group GG is abelian, as we have shown that for any two elements a,bGa, b \in G, we have ab=baab = ba.


Would you like further clarification or a more detailed explanation of any part of the proof?

Related Questions:

  1. What is the definition of an abelian group?
  2. Can a group have more than one element of order 2?
  3. What does it mean for a group element to have order 2?
  4. How do we prove that a group is cyclic?
  5. What are the properties of cyclic groups?

Tip: The key to proving that a group is abelian in such cases is to use the properties of group elements, such as their orders and inverses, to manipulate the group's structure algebraically.

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Math Problem Analysis

Mathematical Concepts

Group Theory
Abelian Groups
Order of Elements

Formulas

Order of an element: a^n = e implies the order of a is n
(ab)^2 = e implies ab = ba for groups where all elements have order 2

Theorems

Basic properties of groups
Definition of an abelian group

Suitable Grade Level

Undergraduate Mathematics or Advanced High School (Grades 11-12)