Gracious Living

More Products of Groups
November 27, 2010, 22:40
Filed under: Algebra, Math | Tags: , , , ,

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 E(n) 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 {\rm Aut}(X), the automorphism group of X, whose elements are the isomorphisms from X to itself.  Depending on what X is, this sometimes has a different name, like {\rm Homeo}(X) for the group of self-homeomorphisms of a topological space.

The structure of X can also endow {\rm Aut}(X) with additional structure.  In the case of a group G, for example, there is always a homomorphism G\rightarrow{\rm Aut}(G) given by sending g to the automorphism “conjugation by g.”  (As a reminder, this is x\mapsto g^{-1}xg.)  The image of this map is the set of automorphisms that come from elements of G, and is called {\rm Inn}(G), the inner automorphism group.  (We can define the outer automorphism group as {\rm Aut}(G)/{\rm Inn}(G), as well.)

If G is abelian, then g^{-1}xg=xg^{-1}g=x for all g,x, so all inner automorphisms are trivial, and conversely, if {\rm Inn}(G) is trivial, then G is abelian.  We can go even further: define the center of G, Z(G), as the set of elements that commute with every element of G.  The kernel of the map G\rightarrow{\rm Inn}(G) 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, Z(G) is a normal subgroup of G, and G/Z(G)\cong{\rm Inn}(G).  This shows how the structure of {\rm Aut}(G) is determined by G‘s group structure.

Now let’s look at the semidirect product.  Recall that the direct product G\times H has G and H as normal subgroups; likewise, given two normal subgroups G,H of J, with GH=J,G\cap H=\{e\}, we have J\cong G\times H.  A generalization of this exists when only G is normal.  Say we still have GH=J,G\cap H=\{e\}.  By the same reasoning as for the direct product, every element of J is a unique product of an element of G and an element of H.  Likewise, we have GH=\{gh\}=\{(hg^\prime h^{-1})h\}=\{hg^\prime\}=HG, but notice the subgroups no longer commute with each other.  Instead, we have gh=h(h^{-1}gh), i. e. g is conjugated by h when we pull h through it.  So we can write J as the set of elements (g,h),g\in G,h\in H, with (g_1,h_1)(g_2,h_2)=(g_1(h_1g_2h_1^{-1}),h_1h_2).  In this case, we say J is the semidirect product of H acting on G, J=G\rtimes H.  The order of the factors matters here.  I may not have mentioned this before, but G\triangleleft J is a common symbol for “G is a normal subgroup of J, so the triangle in the product symbol indicates that G is the normal one.

The “external” case is similar.  Let G and H be two groups, with a given map \phi:H\rightarrow{\rm Aut}(G), where we’ll write the image of h\in H as \phi_h:G\rightarrow G.  The semidirect product of H acting on G by \phi is G\rtimes_\phi H=\{(g,h)\} with (g_1,h_1)(g_2,h_2)=(g_1\phi_{h_1}(g_2),h_1h_2).  The internal case is given by \phi_h being “conjugation by h^{-1}.”  You might want to check that this being a homomorphism to {\rm Aut}(G) is equivalent to G being normal and G\cap H=\{e\}.  Direct products are also a special case, when \phi_h is the identity map for all H.

The group of Euclidean isometries is really the ur-example, but there are a few others.   For example, the dihedral group D_{2n} is a semidirect product of \mathbb{Z}_n and \mathbb{Z}_2, where 1\in\mathbb{Z}_2 sends k\in\mathbb{Z}_n to n-k.  The orthogonal group O(n) is a semidirect product of \mathbb{Z}_2 and SO(n), where again the nonzero element of \mathbb{Z}_2 inverts rotations.  Since the underlying set of G\rtimes H is just G\times H, we have |G\rtimes H|=|G||H|.

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 G and H, we define G*H as the set of reduced words in elements of g and h, 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 g_1h_1g_2h_2\dotsm g_nh_n).  Multiplication is just what you’d expect: concatenate the two words, then reduce.  In generators-and-relations format, if G=\langle Q_G|R_G\rangle,H=\langle Q_H|R_H\rangle, where the Qs are generators and the Rs are relations, then G*H=\langle Q_G\sqcup Q_H|R_G\sqcup R_H\rangle.

Once again, we have canonical injections that just send each element of G or H to the corresponding one-character word.  If G and H map into another group, we just apply more relations to get a map from G*H to that group.  For example, the maps G,H\rightarrow G\times H factor through G,H\rightarrow G*H via a map i:G*H\rightarrow G\times H with \ker(i)=\{ghg^{-1}h^{-1}:g\in G,h\in H\}.  (This is just forcing elements of G to commute with elements of H.)  Neither G or H are normal in G*H unless one is trivial, and |G*H| is infinite unless one of the groups is trivial.  And they call themselves “free.”  Oh, but F_n is the free product of n copies of \mathbb{Z}.

A sort of funky generalization of this occurs when we have maps a:F\rightarrow G,b:F\rightarrow H for some group F.  The free product of G and H amalgamated over F is G*H with the extra relations a(f)b(f)^{-1} for each f\in F.  This is precisely (G*H)/N, where N is the smallest normal subgroup of G*H containing all those relations.  We sometimes write this as G*_FH.  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 G_i\rightarrow\prod_i G_i don’t generate \prod_i G_i — they only generate the set of elements of whom all but finitely many coordinates are the identity.  This group is called the direct sum \bigoplus_i G_i.  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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