Group can be determined by how elements multiply with each other

In summary: Associativity is just saying that the order of operations doesn't matter- that every operation is the same no matter what order you do them in. Identity and inverse elements are straightforward- you just need an element that behaves like every other element, and that element is 1. The last two axioms just say that if a and b are elements in G, and ab is calculated, then ba is also calculated.
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tgt
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Given a group, it seems everything (all properties) about the group can be determined by how elements multiply with each other. Correct? For example, everything about the group could be read off from the multiplication table.
 
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1. Closure. For all a, b in G, the result of the operation ab is also in G.
2. Associativity. For all a, b and c in G, the equation (ab)c = a(bc) holds.
3. Identity element. There exists an element 1 in G, such that for all elements a in G, the equation 1a = a1 = a holds.
4. Inverse element. For each a in G, there exists an element b in G such that ab = ba = 1, where 1 is the identity element.

After looking at the axioms one at a time, it sure looks like they all can be satisfied for a finite group in a multiplication table.
 
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well it is hard to look at the multiplication table of a group of order 60 and decide whether it is simple or not, but in principle yes, everything is determined by the multiplication table.

but that's like saying everything about a solid is determined by its equation, but you still may have trouble computing its volume from that knowledge. e.g. the length of the graph of y = x^3 is determined by that equation, but what is that length from x=0 to x=1?
 
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but still, even if you don't have a group of a that big order, say you have a group of 6 elements, then it would be a pain to check the associativity from the operation table. since there would be x nr of cases, where x is the nr of permutations with repitition.
 
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tgt said:
Given a group, it seems everything (all properties) about the group can be determined by how elements multiply with each other. Correct? For example, everything about the group could be read off from the multiplication table.
Well, yeah! A group is defined by its members and its operation- there is nothing else. Of course that operation is not always defined in terms of a multiplication table, but you could, theoretically, write one. (That might be very difficult in the case of infinite groups!)
 

FAQ: Group can be determined by how elements multiply with each other

What is a group in mathematics?

A group in mathematics is a set of elements that satisfy certain properties, such as closure, associativity, identity, and inverse. These properties allow for operations, such as multiplication, to be performed within the group.

How is a group determined by how elements multiply with each other?

A group is determined by how elements multiply with each other because the operation of multiplication must follow the properties of a group. This means that for any two elements in the group, their product must also be in the group and the product must be associative, have an identity element, and have an inverse element.

What is the difference between a group and a subgroup?

A subgroup is a subset of a group that also satisfies the properties of a group. This means that all elements in the subgroup must also be in the larger group and the operation of multiplication must follow the properties of a group. However, a subgroup may have fewer elements and a different identity element than the larger group.

Can there be multiple groups with the same elements?

Yes, there can be multiple groups with the same elements as long as the operation of multiplication follows the properties of a group. This means that the same set of elements can form different groups depending on the operation performed on them.

How are groups used in mathematics and other fields?

Groups are used in mathematics to study symmetry, transformations, and other abstract concepts. They are also used in other fields, such as physics and chemistry, to describe and analyze the behavior of systems and structures. Groups can also be applied to real-world problems in fields like computer science and cryptography.

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