13.3. Examples of Groups

So now, let us see some examples of groups. So, the set of integers ℤ which is an infinite set along with the operation + constitutes a group. So, let us see whether all the 4 properties are satisfied or not. So, the closure property is of course satisfied; you take any 2 integers  and  and add them you will again obtain an integer. The operation is associative over the integers too since if you take any 3 integers, it does not matter in what order you add them, the result will be the same. The integer 0 is the identity element  because adding 0 to any integer  will result in the same integer . And the integer − will be considered as the inverse of the integer . So, this − is actually  as per the notation and you can see that you take any integer , its inverse is − because if you add − to  then the result will be 0, which is the identity element.

In this chunk, an example of a group is provided using the set of integers combined with addition as the operation. To determine if this indeed forms a group, we check four properties: closure, associativity, identity, and inverse. The closure property states that adding any two integers results in another integer, which is true. The associativity property indicates that the order of addition does not affect the sum. The integer 0 acts as the identity element, as adding 0 to any integer does not change its value. Finally, for each integer, there exists an additive inverse (e.g., the inverse of integer 5 is -5) that leads back to the identity element when added.

Think about a bag of apples you collect. Whether you add two apples (2 + 3) to your bag or three apples first and then two (3 + 2), you still know you have five apples in your bag (associativity). Adding zero apples means your total remains unchanged. For each apple in your bag, if you imagine taking it away (like having an inverse), you still can return to zero apples (the identity).

Here the set  was the set of integers. Now, let  be the set of non-negative integers ℤ i.e., negative integers are not included. The operation is still the same, namely +. Now, it is easy to see that this  along with the + operation does not constitute a group. Which property is violated? Here the closure and associative properties are still satisfied and the identity element 0 is still present in . The issue is that the fourth group axiom is not satisfied, because the inverse of an integer  will be −, but − is not an element of ℤ because − is a negative integer.. Whereas the group axiom says that the inverse element also needs to be a member of the set  itself. So, that is why the set of non-negative integers along with the addition operation does not constitute a group.

In this chunk, we analyze the set of non-negative integers and see if it forms a group under addition. Although closure, associativity, and the presence of an identity (0) are confirmed, it fails the inverse property. Specifically, for any non-negative integer, its additive inverse (like -5 for 5) is not a part of the set of non-negative integers. This violates the group's requirement that every element must have an inverse that is also in the set, indicating that this collection does not fit the definition of a group.

Imagine you have a collection of only ripe apples (representing the non-negative integers). Although if you add apples together, you always get more ripe apples (closure), and if you don't add any apples, you still have the same amount (identity), you can’t take away apples and still have ripe ones left (inverse). If you pick an under-ripe apple (negative), it simply doesn’t fit in your collection of ripe apples!

So, we have seen now a group with respect to the + operation now let us see a group with respect to the multiplication operation. So, now my set  is the set of all real numbers excluding 0 and my operation ∘ is the multiplication operation and now you can see that all the 4 properties of group are satisfied. Multiplying any two real numbers will give you a real number and multiplication is associative over the real numbers. The real number 1 is the identity element because you multiply 1 with any non-zero real number , you will obtain the same non-zero real number . And you take any non-zero real number , its multiplicative inverse will be 1/. And it is well-defined because 1/ is non-zero. So, indeed it exists and it belongs to the set of non-zero real numbers. So all my 4 group axioms are satisfied and hence this set constitutes a group.

In this chunk, we explore the set of all non-zero real numbers under multiplication. Here, we find that all four group properties hold true: closure (the product of two non-zero real numbers is a non-zero real number), associativity (the order of multiplication does not matter), identity as 1 (since multiplying by 1 returns the original number), and the existence of an inverse (the reciprocal of any non-zero number is also non-zero). Therefore, this structure indeed qualifies as a group.

Consider a culinary recipe: if you multiply the ingredient amounts (like 1 cup of flour) by 1, the amount stays the same (identity). Even if you combine flour with another ingredient (2 cups of rice), you still know how much you have in total. For any ingredient, if you know how much you’ve used, you can easily calculate how much you have left by dividing (like finding the multiplicative inverse).

Now let us see some other interesting examples of groups. So let k be a positive integer and let ℤk be the set of integers 0 to k−1. Basically, it is the set of all possible remainders which you can obtain by dividing any integer, it could be either positive or negative, by k. Now I define a new form of addition over this set called addition modulo k, which is defined as follows: I add a and b and then take modulo k, the result will be called as the result of addition of a and b modulo k i.e., a +b = [a + b] mod k. So, now my claim is that this set ℤk, which is a finite set because k is a positive integer, constitutes a group with respect to this operation of addition modulo k.

This chunk introduces a new example using a finite set defined by integers from 0 to k-1 and a unique addition defined as addition modulo k. This operation takes the normal sum of two numbers, then finds the remainder after division by k. We assert this forms a group, which can be shown by validating the four axioms of groups: closure (the sum will always fall within the defined set from 0 to k-1), associativity (it doesn't matter the order of operations), identity (adding 0 leaves the number unchanged), and an inverse (each number can find a complement that returns to 0 when added together).

Imagine you have a clock set to a maximum of k hours. If it's 9 o'clock and you add 5 hours, logically it would show 2 o'clock again (modular addition). Everyone understands that each hour behaves like the remainder of division by 12 on a clock, demonstrating closure and the cyclical nature of time (associativity). Adding 0 hours leads to the same time, and the inverse of 2 o’clock is 10 o'clock (since that will return to the start of the cycle).

Now let us see a corresponding variation of the multiplication operation, which we call as multiplication modulo k. So, let ℤ∗ be the set of all integers a in the set ℤ which are co-prime to the modulus k. Now, we define a new operation called multiplication modulo k which is a variation of multiplication over the elements of ℤ∗. So, my claim is that this set ℤ∗ with respect to this operation of multiplication modulo k constitutes a group. So, let us see whether the closure property is satisfied or not.

This chunk examines a further example of a group formed by the set of integers co-prime to a modulus under multiplication modulo k. The definition of co-prime means that an integer has no common factors with k apart from 1. We claim this set forms a group, and we start testing the group properties such as closure (multiplying two co-prime numbers modulo k should also yield another co-prime number) and others like identity, associativity, and the existence of inverses similar to previous examples. Careful proofs ensure the necessary properties hold.

Imagine you have a selection of colorful beads for jewelry making, where only certain colors pair well (co-prime) without clashing (no common factors). When you combine these beads (multiply), you end up with a new bead color that also works within the design (closure). The unique colors behave consistently, showcasing the reliable nature of your crafting options (associativity). Adding a clear bead (identity) doesn’t change your mix, and every bead has a pairing that matches back to your original design.

So, now we have seen examples of several groups. Namely we have seen examples of 4 groups, each group has a different structure; namely their elements were different and the operations were also different. And these are not the only examples of groups, I can give you infinitely many examples.

In this conclusion chunk, a summary of the diverse group examples discussed is provided. Each group discussed has a unique structure dependent on the elements included and the operations defined. Additionally, it is acknowledged that these examples are merely a subset of the many possible groups that can exist and satisfy the group axioms. This example variety highlights the versatility of group theory in abstract algebra.

Think of different genres of music—each genre (like rock, jazz, classical) has its unique elements and styles (akin to groups), yet they all still belong to a broader field of music theory (similar to group theory). Just as there are countless musical compositions that can exist within these genres, there are endless examples of groups in mathematics.

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