Testing whether a binary structure is a group

In summary, the binary structure given by multiplication mod 20 on {4, 8, 12, 16} is a group, as it is isomorphic to the Klein four-group. This is sufficient to show that the structure is associative, as the isomorphism implies that the other group is associative. Additionally, the closure of the operation within the set is also guaranteed.
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Mr Davis 97
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Homework Statement


Consider the binary structure given by multiplication mod 20 on {4, 8, 12, 16}.
Is this a group? If not, why not?

Homework Equations

The Attempt at a Solution


I started by constructing a Cayley table, and working things out. It turns out that 16 acts as an identity element, 4 is the inverse of itself, 12 and 8 are mutual inverses, and 16 is the inverse of itself. One more things to check would be to see if the associative property is satisfied for all elements. However, this would seem to be a very tedious process.

On the other hand, I know that, up to isomorphism, there are only two types of groups of order 4, the cyclic group ##\mathbb{Z}_4## and the Klein four-group. Just by comparing tables, the Cayley table for this binary structure is equivalent to that of the Klein four-group. So is it valid to say, by isomorphism, that this binary structure is also a group, or do I have to explicitly show associativity?
 
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The isomorphism is sufficient, because you already know, that the other group is associative. Beside that, you only need the closure, i.e. that the multiplication stays inside the set. Associativity is then inherited by ##\mathbb{Z}_{20}## or even by ##\mathbb{Z}## itself.
 
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1. What is a binary structure?

A binary structure is a set of elements with a binary operation defined on it. The binary operation takes two elements from the set and combines them to produce a new element in the set.

2. What is a group?

A group is a type of binary structure that satisfies four properties: closure, associativity, identity, and inverses. Closure means that the result of the binary operation on any two elements in the group is also in the group. Associativity means that the order of operations does not matter. Identity means that there is an element in the group that does not change when operated on by any other element. Inverses means that for every element in the group, there is another element that, when operated on together, produces the identity element.

3. How do you test if a binary structure is a group?

To test if a binary structure is a group, you must check if it satisfies the four properties of a group: closure, associativity, identity, and inverses. This can be done by performing various operations on the elements in the structure and checking if the results follow these properties.

4. What is the significance of testing if a binary structure is a group?

Testing if a binary structure is a group is important in mathematics and science, as it allows us to understand the properties and behavior of different structures. In particular, groups are used in abstract algebra to study symmetry and transformations, and in computer science to design efficient algorithms and data structures.

5. Are there any real-world applications of testing if a binary structure is a group?

Yes, there are many real-world applications of testing if a binary structure is a group. For example, in chemistry, the concept of a group is used to understand the symmetries of molecules and their behavior. In physics, groups are used to model symmetries in space and time. In computer science, groups are used to design data structures and algorithms for efficient processing of large amounts of data. In cryptography, groups are used to design secure communication protocols.

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