As I was preparing the pages summarizing some definitions and facts from algebra, I accumulated some stuff that’s interesting, or may be useful to have for reference, but that we won’t be using in this course, and you don’t need to master or remember. So regard all of the following as optional. Look at it if you like, but do not let it overwhelm you. Maybe some of it will be helpful to you someday.
Some more remarks about permutations.
Α
(see Homework Problems 13 and 51), is the symmetric group on Α
.G
is isomorphic to a group of permutations of G
’s universe. 3 ↦ 12, 12 ↦ 20, 20 ↦ ... ↦ 5, 5 ↦ 3
. (All elements not included in a cycle should be understood to map to themselves.)The center of a group G
with operation ★
is the set of elements that commute with every other element: {g ∈ G | ∀ x ∈ G (g ★ x = x ★ g)}
. This will always include the group’s identity, and if ★
is commutative, will include the group’s whole universe.
We explained the notion of a subalgebra, and gave a subgroup as one example. The notion of a subring is defined analogously.
Some elements a,b,c,...
are said to generate a group ⟨Α, ★, Ⓐ⟩
if every element of the group is equal to one of those elements, or their inverse, or the result of applying ★
to several instances of the foregoing. When a group can be generated by a single element, it’s called a cyclic group. Often the choice of the generating element is not uniquely constrained: for example, ℤ₅
could be said to be generated by 1
or by 2
(or indeed by any of its non-0
elements).
Some interesting facts about cyclic groups:
Sometimes the term order is used to talk about the size of a group’s universe. But this term is also used in another way, to talk about a property of elements of a group rather than the group itself. If ⟨Α, ★, Ⓐ⟩
is a group, the order of an element a ∈ Α
in that group is the smallest positive n
such that a ★ a ★ ... ★ a = Ⓐ
, with n
instances of a
. If there is no such n
, then a
is said to have infinite order in that group. These two uses of the term “order” are connected, in that for cyclic groups, the order/size of the group is provably equal to the order of the group’s generating element.
Additionally, cyclic groups of order n
are always isomorphic to ℤ_{n} (we can understand ℤ_{∞} to be just ℤ
).
Let’s say that H
is a subgroup of G
, both having operator ★
. Then the left cosets of H
in its parent group G
are subsets of G
’s universe, defined for a given choice of g ∈ G
: specifically, the sets {g ★ h | h ∈ H}
. These can be written as g ★ H
. (This doesn’t mean that the group H
is a legitimate argument of the ★
operation; it’s shorthand for talking about the set you get by taking elements from H
and applying g ★
to them.)
So for example, if {Ⓖ, g, g′, g″, ...}
is G
’s universe, then Ⓖ ★ H
, g ★ H
, g′ ★ H
, g″ ★ H
, will be left cosets of H
in G
. If Ⓖ
is the identity for ★
, then the first of those will be H
’s universe. Some of the cosets may be the same set, and in particular if g′ ∈ g ★ H
then g′ ★ H = g ★ H
. In other words, the cosets don’t overlap, and so the set of all of them will partition the universe of G
. Moreover, each coset will have the same size.
The index of H
in G
, written (G : H)
, is the number of cosets of H
in G
.
The notion of a right coset is defined similar to the above. When the operations we’re working with aren’t commutative, we can’t in general rely on the left and right cosets to be the same; but there will be the same number of them and they’ll be structurally alike.
In the same way, we can define the notion of cosets of a subring in its parent ring. The set of all those cosets will also partition the universe of the parent ring.
We described a (homo)morphism f
from the group ℤ₆
to the group ℤ₃
, defined as follows:
f(0) = 0
f(1) = 1
f(2) = 2
f(3) = 0
f(4) = 1
f(5) = 2
The kernel of a (homo)morphism is the set of elements in the source algebra that it maps onto the identity element in the target. So if the example just given, the kernel would be {0,3}
.
Again, let H
be a subgroup of G
, both having operator ★
. An interesting special class of subgroups have the property that for all g ∈ G
and h ∈ H
, the element g ★ h ★ g̅
is also ∈ H
. Such elements are called conjugates of h
, and when a subgroup H
is closed wrt conjugates in this way, it’s called a normal subgroup of G
. There is an analogous notion for rings, which we might think of as “normal subrings,” but these are usually called by the special name ideals.
There are important relations between these notions and the notion of the kernel of a (homo)morphism, explained just above. The first of these important facts are that, if f
is a (homo)morphism between groups, then its kernel is a normal subgroup of the original group. Similarly, if f
is a (homo)morphism between rings, then its kernel is an ideal/normal subring of the original ring.
The second of these important facts brings in the notion of a coset, explained before. Since the kernel of an (homo)morphism is a subgroup, we can talk about the set of its cosets (either left or right, with normal subgroups these turn out to be the same). We said before that this set of cosets will partition the universe of the parent group. Indeed, it can be shown that the (homo)morphism whose kernel we’re talking about maps every member of the same coset to the same element in the target group. The upshot of this is that we can construct a group whose universe is the set of all cosets of a given kernel K
, and whose operation ⊛
can be defined so that (K ★ a) ⊛ (K ★ b) = K ★ (a ★ b)
. This group is called the quotient group of G
by K
, and is written G/K
.
The term “quotient” is used because the group G/K
will tend to lack interesting properties had by the members of K
; so this transformation is a way of “dividing away” those properties from G
, and getting a group with only those parts of its structure that remain.
As an example, let 6ℤ
be the cyclic subgroup of ℤ
generated by 6
, that is, the subgroup whose universe is {..., -6, 0, 6, 12, ...}
. Then the quotient group ℤ/6ℤ
will be a group whose universe includes as an element the set just mentioned, whose members mod 6
are all 0
, as well as the set of integers whose members mod 6
are all 1
, and so on. This group is isomorphic to the group we were calling ℤ₆
.
The notion of a quotient ring can be defined similarly.
The third important fact is that not only can these quotient groups and rings be defined, but every group homomorphic to G
turns out to be isomorphic to the quotient group G/K
, where K
is the (homo)morphism’s kernel. So too for rings and quotient rings.
We mentioned the notion of an integral domain, which is an algebraic structure “in between” rings and the more specific fields. Here are some more comments about that.
★
. Because of this, it will have a general property of groups that I’ll repeat here. For any elements w,x,y
:Whenever x ★ w = y ★ w
, x = y
.
Whenever w ★ x = w ★ y
, x = y
.
We can call this the “cancelation property” wrt ★
. Some rings are such that they also have the cancelation property wrt ⊙
.
A different property some rings have is that there are elements d,e
distinct from ⓪
(the ★
-identity) such that d ⊙ e = ⓪
. For instance, recall that ℤ₆
can be regarded not merely as a group (with the operation of addition mod 6
), but also as a ring, where the ⊙
operation is multiplication mod 6
. In this ring, 2 ⊙ 3 = 0
. Such elements, when they exist, are called “divisors of zero.”
An interesting fact about rings is that they have the cancelation property wrt ⊙
iff they have no divisors of zero. (So for rings with the cancelation property, d ⊙ e = ⓪
implies d = ⓪
or e = ⓪
.)
Integral domains are specifically those rings that have a commutative ⊙
with an identity ①
, and that have the cancelation property wrt ⊙
.
As I said in the main discussion, the paradigm example of this are the integers, that is the algebra ⟨ℤ, *, 1, +, 0⟩
. Here I list the identity for *
as well as the identity for +
. But there are also other examples, such as ⟨ℚ, *, 1, +, 0⟩
. If we lay down further constraints on the integral domain having to do with how its elements can be ordered, then that will exclude the algebra ⟨ℚ, *, 1, +, 0⟩
and every other algebra that isn’t at least isomorphic to the integers.
The integers aren’t a field, because they don’t include enough *
-inverses. But every integral domain with a finite universe turns out to be a field.
An ideal/normal subring of a ring is called maximal if it’s the largest ideal that’s still smaller than the original ring. It turns out that if R
is a ring whose ⊙
operation is commutative and has an identity, then a subring J
will be a maximal ideal iff R/J
is a field.
In any such ring R
, the principal ideal generated by an element r
is the subring whose universe is the coset r ⊙ R
. (In some rings, this will be identical to the original R
.)
An element a
of a ring ⟨Α, ⊙, ★, ⓪⟩
is called nilpotent when a ⊙ a ⊙ ... ⊙ a = ⓪
, for some positive number of instances of a
. If the ring has a ⊙
-identity ①
, then an element b
is called unipotent when ① ★ b̅
is nilpotent.
We said before that ⟨ℤ₄, * mod 4, + mod 4, 0⟩
is a ring where *
has identity 1
. In this ring, 0
and 2
are nilpotent element, and 1
and 3
are unipotent element.