Is <Z_n\{0},+> a Valid Group?

In summary, The group <Z_n\{0},+> does not contain the identity element and is not closed under addition, therefore it is not a group.
  • #1
nille40
34
0
Hi!
Let us say we have a group [tex]\langle Z_n \backslash \lbrace 0 \rbrace, \cdot \rangle[\tex] and one element multiplied with another gives kn. n divides kn, so kn equals 0. But we don't have 0 in the set of the group.

If the tex stuff didn't show up, the group should be
<Z_n\{0},+>

Is this then not a group?
Nille
 
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  • #2
Originally posted by nille40


Is this then not a group?
Nille

You re damn straight its not a group! a group contains the identity! remove the identity, and you no longer have a group on your hands... but why would you want to go and do something so perverse like remove the identity?
 
  • #3
Even worse, it's not closed under addition. 1+(-1)=0, which is not an element in the given set.
 

1. What is a finite group?

A finite group is a mathematical structure that consists of a set of elements and a binary operation that combines any two elements to form a third element. The set of elements in a finite group is finite, meaning it has a finite number of elements.

2. What are the properties of a finite group?

A finite group must satisfy four properties: closure, associativity, identity, and inverse. This means that the binary operation must always result in an element of the group, the order in which the operation is performed does not matter, there is an element that acts as an identity under the operation, and every element has an inverse that when combined with the element under the operation gives the identity element.

3. How is a finite group different from an infinite group?

The main difference between a finite group and an infinite group is the number of elements in their sets. A finite group has a finite number of elements, while an infinite group has an infinite number of elements. Additionally, finite groups have different properties and behaviors compared to infinite groups.

4. What are some examples of finite groups?

Some common examples of finite groups include the group of integers modulo n under addition, the symmetric group on n letters, and the dihedral group of order n. The Klein four-group, cyclic groups, and permutation groups are also examples of finite groups.

5. How are finite groups used in science?

Finite groups are used in various fields of science, including physics, chemistry, and computer science. In physics, finite groups are used to describe the symmetries of physical systems, such as in crystal structures. In chemistry, finite groups are used to study molecular structures and reactions. In computer science, finite groups are used in cryptography and coding theory for secure communication and data storage.

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