SOLUTION: Let G be a group such that x^2 = e for all x ∈ G. Show that G is abelian.

SOLUTION: Let G be a group such that x^2 = e for all x ∈ G. Show that G is abelian.

Algebra.Com

Question 1018437: Let G be a group such that x^2 = e for all x ∈ G. Show that G is abelian.
Answer by richard1234(7193) (Show Source): You can put this solution on YOUR website!
For all

, we have

since x*y produces another element in the group.

If we consider the expression

, note that y*y = e, so the expression reduces to

. So

for all

However since

and

are both inverses of x*y in G, we must have that

for all x,y in G.
RELATED QUESTIONS
Let g be an element of a group (G, ∗) such that for some one element x ∈ G, (answered by richard1234)

G is a finite semigroup such that for each x, y, z, if xy = yz, then x = z. Prove that G (answered by robertb)

b) Let g be a function such that g^-1 exists for all real numbers. Given that g(9) = 4, (answered by Solver92311)

Let g be a linear function such that g(-1)= -3 and g(-4)= 12 Find an equation for g(x) (answered by josmiceli)

Let g be a function such that g^-1 exists for all real numbers. Given that g(9) = 4,... (answered by ikleyn)

For f(x)=(3x^2)+5and g(x)=7x-2, a. Verify: g(x + 2) ≠ g(x) + g(2). b. Find... (answered by MathLover1)

Let f(x)=x^3+2x^2-3x-1. Write a function g such that... (answered by LinnW)

Let g and h be elements of a group G. show |(g)*(h)*(g(inverse))| =... (answered by venugopalramana)

Given h(x) = cos sqrt(3x^2-7), determine f(x) and g(x) in two different ways, such that... (answered by greenestamps)