Every group1 has an intrinsic group, its automorphism2 group, which is the set of the group's automorphisms under function composition. At first, I thought the automorphism group was every permutation of the set of the elements in the original group. Let’s see why this is not so, and let's see what characteristics the automorphism group reveals about group elements.
Say you have Z4, integer addition in modulo 4. Can't you just create any bijection from Z4 to Z4 and call it an automorphism? Such as
Z4->f->Z4
f(x) = x+1 mod4 ?
No. Because this bijection does not obey the property of homomorphisms.
f(x+y) =/= f(x)+f(y).
In the mapping
f(x) = x + 1 mod4
if we take f(2+2), we get
f(4) = 4+1 mod4 ~= 1.
But
f(2)+f(2) = (2+1)+(2+1) mod4 ~= 2.
Seeing which maps are valid and which maps are invalid reveals to us that elements in a group can be categorized based on what elements they can be mapped to in an automorphism.
In Z4, 2 has the property of being its own inverse. While 3 and 1 generate3 the group. So the only automorphism that I can come up with (other than the trivial f(x)=x) is swapping 3 and 1, i.e.
f(3) = 1 and f(1) = 3, and f(0) = 0 and f(2) = 2.
I hypothesize that generators must map to generators and non-generators must map to non-generators in an automorphism. Identity elements always map to themselves.4 The next question that arises: are there other types of elements that must map to elements of that same type? For example, certain elements generate a subset of their group and I guess that in an automorphism they need to map to an element in the subgroup that they generate. Or maybe non generators must not map to anything at all. Stay tuned!
- A group is a set along with a binary function that takes two elements in the set and produces a third in the same set.
The binary operation is associative, i.e.
(a+b) + c = a + (b + c)
but not necessarily commutative. If it is commutative, the group is an abelian group. Each group has an additive identity, i.e. a 0 element, where
a + 0 = a for all a in G
and each element has an inverse, where
a + a-1 = 0.
An example of a group is the integers under addition. An example of a non-group is integers under multiplication - all elements, except 1, lack an inverse without expanding the set to the rationals. [↩]
- An automorphism is a type of isomorphism, which is a type of homomorphism. A homomorphism, f(x), of groups is a mapping that takes an element in one group, G1, to another group, G2 such that
f(xG1+yG1) = f(xG1)+f(yG1)
An isomorphism is a homomorphism that is a bijection. Homomorphisms don't require a full image.
An automorphism is an isomorphism between a group and itself.
After learning these definitions I was confused as to the difference between groups that are isomorphic and a group’s automorphisms. My understanding is that a group has an infinite number of groups isomorphic to itself. This is because you can define a new group by simply changing the name of the elements inside the group. So once you understand the properties of a group, you obviously understand all of the group's isomorphisms. [↩]
- An element g generates a group whenever repeated addition to itself gets you a cycle that goes through every other element in the group, i.e.
g generates G iff x = gn for some n for all x in G.
Note also that raising to the power of n in the context of a group signifies repeating the binary operation on n and does not necessarily mean repeated "multiplication" as defined in elementary arithmetic. [↩]
- Let's prove this real quick.
f(a + e) = f(a)
and
f(a+e) = f(a) + f(e)
for all a. So for all a,
f(a) + f(e) = f(a),
so if f is a bijection than f(e) is the identity element in the codomain of f. [↩]