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Group theory is the study of groups. Groups are sets equipped with an operation (like multiplication, addition, or composition) that satisfies certain basic properties. As the building blocks of abstract algebra, groups are so general and fundamental that they arise in nearly every branch of mathematics and the sciences.
- Sets
- Operations
- Binary Operations
- Well Defined
- Introduction to Groups
- Formal Definition of A Group
- Only Two Operations
- Why Groups?
- Special Types of Groups: Abelian
Before reading this page, please read Introduction to Sets, so you are familiar with things like this: 1. Set of clothes: {hat, shirt, jacket, pants, ...} 2. Set of even numbers: {..., -4, -2, 0, 2, 4, ...} 3. Positive multiples of 3 that are less than 10: {3, 6, 9}
Now that we have elements of sets it is nice to do things with them. Specifically, we wish to combine themin some way. This is what an operation is used for. An operation takes elements of a set, combines them in some way, and produces another element. or, more simply: An operationcombines members of a set.
So far we have been a little bit too general. So we will now be a little bit more specific. A binary operation is just like an operation, except that it takes 2 elements, no more, no less, and combines them into one. You already know a few binary operators, even though you may not know that you know them: 1. 5 + 3 = 8 2. 4 × 3 = 12 3. 4 − 4 = 0 The...
One thing about operators is that they must be well defined. But reverse that. They must be defined well. Think about applying those two words, "defined well" to the English language. If a word is defined well, you know exactly what I mean when I say it. 1. The word "angry" is defined pretty well, as you know exactly what I mean when I say it. 2. B...
Now that we understand sets and operators, you know the basic building blocks that make up groups. Simply put: A group is a set combined with an operation So for example, the set of integers with addition. But it is a bit more complicated than that. We can't say much if we just know there is a set and an operator. What more could we describe? We ne...
Let's look at those one at a time: 1. The group contains an identity.If we use the operation on any element and the identity, we will get that element back. For the integers and addition, the identity is "0". Because 5+0 = 5 and 0+5 = 5 In other words it leaves other elements unchanged when combined with them. There is only one identity element for...
Way back near the top, I showed you the four different operators that we use with the numbers we are used to: But in reality, there are only two operations! When we subtract numbers, we say "a minus b" because it's short. But what we really mean is "a plus the additive inverseof b". The minus sign really just means add the additive inverse. But it ...
So why do we care about these groups? Well, that's a hard question to answer. Not because there isn't a good one, but because the applications of groups are very advanced. For example, they are used on your credit cards to make sure the numbers scanned are correct. They are used by space probes so that if data is misread, it can be corrected. They ...
If a * e = a, doesn't that mean that e * a = a? And similarly, if a * b = e, doesn't that mean that b * a = e? Well, as a matter of fact, it does. But we are careful here because in general, it is not true that a * b = b * a. But when it is true that a * b = b * a for all a and b in the group, then we call that group an abeliangroup. That fact is t...
Mar 13, 2022 · University of South Florida. Definition 2.1: Groups. A group is an ordered pair (G, ∗) where G is a set and ∗ is a binary operation on G satisfying the following properties. x ∗ (y ∗ z) = (x ∗ y) ∗ z for all x, y, z in G . There is an element e ∈ G satisfying e ∗ x = x and x ∗ e = x for all x in G .
This course explores group theory at the university level, but is uniquely motivated through symmetries, applications, and challenging problems. For example, before diving into the technical axioms, we'll explore their motivation through geometric symmetries.
