Filed under: Algebra, Math | Tags: abelian groups, algebra, categorical, group theory, Math

Before looking at solvability and group classification, I want to mention a couple more ways of “building” groups. We’ve already seen how to find subgroups, and how to take the quotient by a normal subgroup, and how to find the direct product of a family of groups. Dual to the direct product is the free product, which generalizes the idea of a free group. The amalgamated free product is just a free product that we neutralize on the image of some map. Also, though the only really good example is the group of Euclidean isometries, the semidirect product is worth a more formal look. Finally, though it’s mostly terminology, I define the direct sum, which is useful for studying abelian groups.

I’m going to begin by describing **automorphism groups**. Given *any mathematical structure*, its set of isomorphisms to itself, also called **automorphisms**, has the defining properties of a group, by “definition” of isomorphism. We want the identity map to be an isomorphism, the composition of isomorphisms to be an isomorphism, and all isomorphisms to be reversible (by other isomorphisms). Thus, we can form the group , the automorphism group of , whose elements are the isomorphisms from to itself. Depending on what is, this sometimes has a different name, like for the group of self-homeomorphisms of a topological space.

The structure of can also endow with additional structure. In the case of a group , for example, there is *always* a homomorphism given by sending to the automorphism “conjugation by .” (As a reminder, this is .) The image of this map is the set of automorphisms that come from elements of , and is called , the **inner automorphism group**. (We can define the **outer automorphism group** as , as well.)

If is abelian, then for all , so all inner automorphisms are trivial, and conversely, if is trivial, then is abelian. We can go even further: define the **center** of , , as the set of elements that commute with every element of . The kernel of the map is then the set of things that act as the identity when they conjugate any other element; this means that they commute with every other element. By the First Isomorphism Theorem, is a normal subgroup of , and . This shows how the structure of is determined by ‘s group structure.

Now let’s look at the semidirect product. Recall that the direct product has and as normal subgroups; likewise, given two normal subgroups of , with , we have . A generalization of this exists when only is normal. Say we still have . By the same reasoning as for the direct product, every element of is a *unique* product of an element of and an element of . Likewise, we have , but notice the subgroups no longer commute with each other. Instead, we have , i. e. is conjugated by when we pull through it. So we can write as the set of elements , with . In this case, we say is the **semidirect product** of acting on , . The order of the factors matters here. I may not have mentioned this before, but is a common symbol for “ is a normal subgroup of , so the triangle in the product symbol indicates that is the normal one.

The “external” case is similar. Let and be two groups, with a given map , where we’ll write the image of as . The semidirect product of acting on by is with . The internal case is given by being “conjugation by .” You might want to check that this being a homomorphism to is equivalent to being normal and . Direct products are also a special case, when is the identity map for all .

The group of Euclidean isometries is really the ur-example, but there are a few others. For example, the dihedral group is a semidirect product of and , where sends to . The orthogonal group is a semidirect product of and , where again the nonzero element of inverts rotations. Since the underlying set of is just , we have .

Dual to the direct product is the **free product**, which generalizes free groups. Besides an important theorem in algebraic topology, this isn’t actually that useful, but I like to find interesting examples of duality where appropriate. Given two groups and , we define as the set of reduced words in elements of and , where a word is now reduced if all adjacent elements in the same group have been multiplied together, and all identity elements cancelled (so that it looks something like ). Multiplication is just what you’d expect: concatenate the two words, then reduce. In generators-and-relations format, if , where the s are generators and the s are relations, then .

Once again, we have canonical injections that just send each element of or to the corresponding one-character word. If and map into another group, we just apply more relations to get a map from to that group. For example, the maps factor through via a map with . (This is just forcing elements of to commute with elements of .) Neither or are normal in unless one is trivial, and is infinite unless one of the groups is trivial. And they call themselves “free.” Oh, but is the free product of copies of .

A sort of funky generalization of this occurs when we have maps for some group . The **free product of and amalgamated over ** is with the extra relations for each . This is precisely , where is the smallest normal subgroup of containing all those relations. We sometimes write this as . The reason I mention this is that it comes up in the same theorem that the free product does, so it’s about as useful.

The final thing to mention is the direct sum. In the post on direct products, I stated that the images of the maps don’t generate — they only generate the set of elements of whom all but finitely many coordinates are the identity. This group is called the **direct sum** . For finite numbers of groups, this is isomorphic to the direct product, but it’s sometimes useful to emphasize that the result is finitely generated. It’s also common in the theory of abelian groups, largely because it’s written with a big giant plus and abelian groups are also usually written additively. The direct sum is the coproduct of abelian groups like the free product is for usual groups: there are canonical injections of each component, and any set of maps to a different group has to factor through the direct sum using those canonical injections. The relation between the two is that the direct sum of abelian groups is the “abelianization” of their free product, the free product mod the set of things that don’t commute.

Thanks for reading!

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