# Determine Whether this is a Group

• cwatki14
In summary, the conversation discusses determining whether the set N={0,1,2,3,...} with the operation * defined as n*m= n+m if n,m are even, -(n+m)-2 if n,m are odd, and m-n-1 if n is odd, m is even is a group or not. The properties of a group, including associativity, existence of an identity, and inversibility, are mentioned. It is noted that the third condition is not satisfied if both n and m are even, leading to the question of whether this is enough to conclude that it is not a group or if other cases should be considered. It is clarified that disproving any essential condition is enough, but there is also

#### cwatki14

I am trying to answer the following question:
Determine whether (N,*) is a group or not. N={0,1,2,3,...} and
n*m= n+m if n,m even, -(n+m)-2 if n,m odd, and m-n-1 if n odd, m even.

I know that the properties of a group are associativity, existence of an identity, and inversiblity. If I consider the case where n,m are both even, clearly the third condition is not satisfied since there does not exist an additive inverse in N. Is this enough to conclude that it is not a group, or do I need to consider the other cases?

hi cwatki14!
cwatki14 said:
Determine whether (N,*) is a group or not.

… the third condition is not satisfied … Is this enough to conclude that it is not a group, or do I need to consider the other cases?

no, disproving any essential condition is enough

(but isn't there also a condition that an identity has to exist, ie n*0 = n ?)

(and anyway N doesn't include negative numbers, so how do the definitions work? )

I think the identity element does exist. The problem statement defines N as including 0 as an element. As far as including the negative numbers, obviously these are not included. I think the definitions just define the operations performed on N in different cases of even, odd properties of 2 elements in N, right? I don't think it is claiming that negative numbers are a part of the set.

but then how can they define …
cwatki14 said:
-(n+m)-2 if n,m odd

… when -(n+m)-2 must be negative, and so can't be in N ?

(and what is n*0 or 0*n if n is odd? )

## What is a group?

A group is a collection of individuals or objects that share common characteristics or goals, and interact with each other in a structured manner.

## What are the defining characteristics of a group?

The defining characteristics of a group include a shared identity, a sense of belonging, a common purpose or goal, and interdependence among its members.

## How do you determine if a group exists?

To determine if a group exists, you can observe if there is a shared identity, a sense of belonging, a common purpose or goal, and interactions among the members that demonstrate interdependence.

## Are all collections of individuals considered groups?

No, not all collections of individuals are considered groups. For a collection to be considered a group, it must have the defining characteristics such as shared identity, sense of belonging, common purpose or goal, and interdependence among its members.

## Can a group exist without a leader?

Yes, a group can exist without a designated leader. In some cases, group members may share leadership roles or make decisions collectively, rather than relying on a single leader.