A brief discussion of sets, subsets and supersets, proper and improper
In math, a set is a collection of objects. If every element of set S is also an element of set T, then S is a subset of T, and conversely, T is a superset of S. Superset is a thing, but the word is not commonly used.
Every set is a subset (and superset) of itself. If there are elements of the superset T that are not in S, then we say S is a proper subset of T.
Here are the symbols of comparison.
T ⊃ S S is a proper subset of T, or T contains S
T ⊇ S S is a subset of T, but not necessarily proper
S ⊂ T S is a proper subset of T, or T contains S
S ⊆ T S is a subset of T, but not necessarily proper
Definition of a subgroup
If H is a subset of group G and H is closed under the group operation, H is a subgroup of G.
Some easy things to say about subgroups.
Every group G is a subset of itself, but not a proper subset.
The identity element e of G is a subset of G, and unless G is a one element group, (e,*) is a proper subset.
A simple group G is a group with more than one element whose only subgroups are G and the identity element. A group having more than one element is usually called non-trivial.
Subgroups of (Z,+)
As was stated in the first post, Z is a group when the operation is addition and the identity element is 0. Z is not a simple group, because nZ is also a group for any integer n in Z. Here are some examples.
6Z = {..., -18, -12, -6, 0, 6, 12, 18, ...}
-7Z = {..., 21, 14, 7, 0, -7, -14, -21, ...}
0Z = {0}
Adding together two multiples of n will give you another multiple of n, so nZ is closed under addition. Addition in these groups is associative because addition in Z is associative.
Practice problems
a) In yesterday's post there was one group introduced that is simple. Find our first example of a simple group.
b) Prove that no group of the form nZ is simple. (The proof has two cases, n = 0 is a different proof from n ≄ 0.)
Answers will be provided tomorrow in the comments of this post.
Tomorrow: The finite cyclic groups known as Zn, where n is an integer greater than 0.
Commentary
Math terminology can be very specific to the field, but math also takes everyday words and gives them new meanings you would never hear outside of a math class. In group theory, there are many examples. We have already defined what the words group, identity, operation, inverse, proper, improper and simple mean in regards to group theory, and we will soon be introduced to the group theory meanings of normal, cyclic, generator, conjugate and others.
We also have some reserved symbols for certain sets of numbers. There are specific, fancy version of capital letters used for integers, rationals, reals and complex number, but the word processor for Blogger doesn't have these symbols available, so I will use letters that are bold and italicized. Here is a snapshot of the official symbols you would find in textbooks.
English speakers will have no problem understanding the selection of R to represent real numbers or C to represent complex, but Q and Z are not as obvious. Q stands for quotient, since every rational can be defined by a fraction, and Z comes from zahlen, the German word for number.
a) The group ({1, -1}, x) is a group of order two, and its only proper subgroup is ({1}, x), so it is simple.
ReplyDeleteb) The two cases of nZ.
case 1, n = 0. This is just ({0}, x), and a one element group cannot be simple by definition.
case 2 n != 0. pick a second integer a, as long as a is not equal to 1, 0 or -1. (anZ, +) is a group and it is a subgroup of nZ.
I'll give a single example here, though no single example is a proof of the infinite possibilities. I do this because the ideas may be new to you an a concrete example may make it easier for you to generalize.
6Z = {..., -24, -18, -12, -6, 0, 6, 12, 18, 24, ...}
12Z = {..., -24, -12, 0, 12, 24, ...}
12Z is a subset of 6Z, and 12Z is a group in its own right, so it is a subgroup of 6Z.