Are All Groups Nonempty and How Does the Group Identity Factor In?

Thread starter blueberryfive
Start date Nov 21, 2011
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Groups

In summary, a group is considered nonempty if it contains at least one element, such as the trivial group which consists of only the identity element. A group cannot be both empty and nonempty, and the consequences of a group being nonempty include having properties such as closure, associativity, and identity. It is necessary for a group to be nonempty in order to have a group structure, as it needs at least one element to serve as the starting point for a defined operation and identity element.

Nov 21, 2011

blueberryfive

Are all groups nonempty? If so, is it because all groups have an identity (element)?

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Nov 22, 2011

Theorem.

Right, the definition of a Group gives us that the group must have a minimum of at least 1 element, the group identity.

Related to Are All Groups Nonempty and How Does the Group Identity Factor In?

1. What does it mean for a group to be nonempty?

A group is considered nonempty if it contains at least one element. This means that the group has a starting point or identity element, and is not void or empty.

2. Are there any groups that are inherently nonempty?

Yes, the trivial group is inherently nonempty. It consists of only the identity element and is the smallest possible group.

3. Can a group be both empty and nonempty?

No, a group cannot be both empty and nonempty at the same time. It is either one or the other.

4. What are the consequences of a group being nonempty?

If a group is nonempty, it means that there is at least one element that can be combined with other elements in the group using the group's operation. This allows for the group to have properties such as closure, associativity, and identity.

5. Is it necessary for a group to be nonempty in order for it to have a group structure?

Yes, a group must have at least one element to have a group structure. Without an element to serve as the starting point, the group cannot have a defined operation or identity element.