# Proving G is a Group: Jim's A-Level P6 Exam Struggle

• Jummeh
In summary, the conversation discusses a tough question about a multiplicative group with a fixed element a satisfying axa = x^-1 for all elements x belonging to G. The conversation involves solving four parts of the question, with the first two parts being solved successfully and the last two parts still remaining unsolved. The conversation also mentions a possible key to solving part three and how result four can be derived from previous rules and the identity of the group.
Jummeh
I was revising for my ALevel P6 exam the other day and I came up against a very tough question...

G is a multiplicative froup with identity element e, and a is a fixed element of G for which axa = x^-1 for all elements x belonging to G. Prove that...

i/ a = a^-1

ii/ ax = (ax)^-1 for all x belong to G //cant find the little symbol

iii/ x = x^-1 for all x belong to G

iv/ xy = yx for all x, y belong to G
I can do parts i and ii

axa = x^-1, let x = e
a^2 = e
a = a^-1

for part ii

axa = x^-1
axax = e
axax(ax)^-1 = (ax)^-1 //ax and (ax)^-1 will cancel
ax = (ax)^-1

but for part iii and iv I seem do always go around in circles or i end up with something obvious like x = x. Unfortunately there aren't any answers to this question.

ps: I hope this was the correct place to post this, i had a look in the homework help section but it was mostly about physics. :s
And i have already taken the exam now but the question is kinda bugging me.

Thanks in advance for any help

Jim

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The result of part two is the key to solving answer three. Maybe if I slightly restate what you've proven, the way to go might become apparent:

Theorem: for all y, (ay) = (ay)^-1

Goal: prove x=x^-1 for all x

And result 4 follows from a few applications of rule 3 and the identity:
(xy)^-1 = (y^-1)(x^-1)

Hi Jim,

Proving that G is a group is definitely a challenging question, especially when it involves multiple parts like this one. I can understand why you may feel stuck or frustrated, but don't worry, let's work through the remaining parts together.

For part iii, we want to prove that x = x^-1 for all x belonging to G. One way to approach this is by using part ii, since we've already shown that ax = (ax)^-1. So, let's start with the left side of the equation:

x = x^-1
Multiply both sides by a:
ax = ax^-1

Now, we can use part ii to rewrite the right side as (ax)^-1, giving us:
ax = (ax)^-1

Since we know that ax = (ax)^-1, we can substitute this into the equation:
ax = ax
And since this is true for all elements x belonging to G, we can say that x = x^-1 for all x belonging to G.

For part iv, we want to prove that xy = yx for all x, y belonging to G. This is known as the commutative property in group theory, and it can be a bit tricky to prove. One approach is to use part iii, since we've already shown that x = x^-1.

xy
Using part iii, we can rewrite this as x^-1y
Now, we can use part ii to rewrite the right side as (x^-1y)^-1:
(x^-1y)^-1
Again, using part iii, we can rewrite this as y^-1x
Finally, using part ii again, we can rewrite this as (y^-1x)^-1, giving us:
(y^-1x)^-1

Now, we can substitute this into the original equation:
xy = (y^-1x)^-1

Since we know that (y^-1x)^-1 = xy, we can say that xy = yx for all x, y belonging to G.

I hope this helps clear up any confusion and gives you a better understanding of proving G is a group. Good luck with your future exams!

## 1. What is a group in mathematics?

A group is a mathematical structure that consists of a set of elements and an operation that combines any two elements of the set to produce a third element. In order for a set to be considered a group, the operation must satisfy four main properties: closure, associativity, identity, and invertibility.

## 2. How can we prove that G is a group?

In order to prove that G is a group, we must show that it satisfies the four main properties of a group: closure, associativity, identity, and invertibility. This can be done by demonstrating that the operation defined on the set of elements follows these properties.

## 3. What is the significance of proving that G is a group?

Proving that G is a group is important because it provides a framework for understanding and solving problems in abstract algebra. Groups have a variety of applications in mathematics, physics, and computer science, making them a fundamental concept in many areas of study.

## 4. What challenges may arise when trying to prove that G is a group?

There are several challenges that may arise when trying to prove that G is a group. One common challenge is ensuring that the operation is well-defined, meaning that it produces a unique result for any two elements in the set. Another challenge may be showing that the operation satisfies the properties of closure, associativity, identity, and invertibility.

## 5. How can understanding groups help with solving problems in mathematics?

Understanding groups can help with solving problems in mathematics by providing a structure that allows for organized and systematic approaches to solving problems. Additionally, the properties of groups can be used to simplify calculations and proofs, making complex problems more manageable.

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