# Thread: How to prove a monoid is a group.

1. ## How to prove a monoid is a group.

Let c be a fixed positive integer, and let * denote the binary operation on the set Z of integers defined be the formula

$x * y = xy + c(x+y) + c^2 - c$

for all integers x, y, and z.

Is (Z, *) a group?

I already proved that it's a monoid. Do I have to do this step by induction? Could someone guide me towards a solution?

2. Originally Posted by pikminman
Let c be a fixed positive integer, and let * denote the binary operation on the set Z of integers defined be the formula

$x * y = xy + c(x+y) + c^2 - c$

for all integers x, y, and z.

Is (Z, *) a group?

I already proved that it's a monoid. Do I have to do this step by induction? Could someone guide me towards a solution?
A group is a monoid where every element has an inverse, $a*a' = e$ where $e$ is your identity and $a$ your element. So, you just plug in you $a$ and $a'$ and see what $a'$ must be for $a*a' = e$.

In these questions always look out for possible illegal steps, such as dividing by zero. Often most elements will have an inverse, but some wont, and the ones which wont will be the ones where zero is in the botton of your inverse...
The text passage contains a mathematical question about proving whether a given monoid is a group. It also provides some guidance on how to approach the problem but does not provide a complete answer.
single_question
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# Thread: How to prove a monoid is a group.

1. ## How to prove a monoid is a group.

Let c be a fixed positive integer, and let * denote the binary operation on the set Z of integers defined be the formula

$x * y = xy + c(x+y) + c^2 - c$

for all integers x, y, and z.

Is (Z, *) a group?

I already proved that it's a monoid. Do I have to do this step by induction? Could someone guide me towards a solution?

2. Originally Posted by pikminman
Let c be a fixed positive integer, and let * denote the binary operation on the set Z of integers defined be the formula

$x * y = xy + c(x+y) + c^2 - c$

for all integers x, y, and z.

Is (Z, *) a group?

I already proved that it's a monoid. Do I have to do this step by induction? Could someone guide me towards a solution?
A group is a monoid where every element has an inverse, $a*a' = e$ where $e$ is your identity and $a$ your element. So, you just plug in you $a$ and $a'$ and see what $a'$ must be for $a*a' = e$.

In these questions always look out for possible illegal steps, such as dividing by zero. Often most elements will have an inverse, but some wont, and the ones which wont will be the ones where zero is in the botton of your inverse...
user
The text passage contains a mathematical question about proving whether a given monoid is a group. It also provides some guidance on how to approach the problem but does not provide a complete answer.

TYPE: SINGLE_QUESTION