# Some clarifications required about the two extremes of general extensions (semi-direct products and central extensions)

My professor made another remark that:

Let's go back to the extension question

$$1 \to A \to G \to B \to 1$$

It turns out that a general extension of $$B$$ by $$A$$ is a mutual generalization of two extremes. One extreme is a semidirect product, in which $$A$$ is complemented but conjugation by $$B$$ (and therefore by elements of $$G$$ in general) can be highly non-trivial. In the other extreme, $$A$$ is a subgroup of the center $$Z(G)$$, so that conjugation in $$G$$ does nothing whatsoever to $$A$$; but the search for a complement of $$A$$ may be an utter failure. This extreme is called a central extension, and it is another construction that you should learn in the context of both Jordan-Holder and p-groups.

Questions:

1. In case of central extensions, why exactly can the search for a complement of $$A$$ be an utter failure?

2. What is meant when he says that conjugation by $$B$$ can be highly non-trivial in case of semi-direct products?

• It's a strange expression to use, but it can be an utter failure because there isn't one to find! If you dig a hole in the group searching for gold that can be an utter failure too. Apr 14, 2020 at 7:46
• @DerekHolt So why can't a complement of $A$ be found, in the case of central extensions?
– user568976
Apr 14, 2020 at 7:56
• As I just said, it cannot be found because there is no complement to find. I don't really understand what you are asking. Apr 14, 2020 at 9:01
• @DerekHolt I'm asking "why is there no complement to find?". Is it a characteristic property of central extensions (that is, complements of subgroups of the center do not exist)? "It cannot be found because there is no complement to find" is a tautology that doesn't really answer this.
– user568976
Apr 14, 2020 at 9:09
• I am afraid I continue to think that your question is equivalent to asking why you cannot find gold if you dig a hole in your garden, but I have written an answer with examples. Apr 14, 2020 at 10:36

I believe that this definition of "central extension", as one with $$A \le Z(G)$$ in which there is no complementis non-standard. The standard meaning of "central extension" is simply one in which $$A \le Z(G)$$, irrespective of whether there are complements are not.

But let me give you three examples in which $$A \le Z(G)$$. In the first of these there is a complement, and in the second and third there is not.

Example 1. Let $$G = \langle x \rangle$$ be cyclic of order 6: so $$x^6=1$$ (the identity). Let $$A = \langle x^3 \rangle$$ be the subgroup of $$G$$ of order $$2$$, and $$B \cong G/A$$ is cyclic of order $$3$$. So $$A \le Z(G)$$: in fact $$G = Z(G)$$. Now let $$C$$ be the subgroup $$\langle x^2 \rangle$$ of $$G$$, which is cyclic of order 3 with $$C \cong G/A$$. Then $$B$$ is a complement to $$A$$ in $$G$$: in fact it is the unique such complement.

Example 2. Now let $$G = \langle x \rangle$$ be cyclic of order 4: so $$x^4=1$$. Let $$A = \langle x^2 \rangle$$ be the subgroup of $$G$$ of order $$2$$, and $$B \cong G/A$$ is cyclic of order $$2$$. Again we have $$A \le Z(G)$$ and $$G = Z(G)$$. This time there is no complement $$C$$ of $$A$$ in $$G$$. Such a complement would have to be isomorphic to $$B$$ (i.e. cyclic of order 2) and satisfy $$A \cap C = \{1\}$$. But $$A$$ is the only subgroup of $$G$$ of order $$2$$, so no such $$C$$ exists.

Example 3. Let $$G = \langle 1,r,r^2,r^3,s,sr,sr^2,sr^3 \}$$ be the dihedral group order $$8$$. That is the group of rotations and reflections of a square, where $$r,r^2,r^3$$ are rotations, and $$s,sr,sr^2,sr^3$$ are reflections. Let $$A=Z(G)$$, which is the subgroup $$\langle r^2 \rangle$$ of order $$2$$. Note that $$B \cong G/A$$ is a Klein four group. A complement $$C$$ of $$A$$ in $$G$$ would be a subgroup of order $$4$$ isomorphic to $$B$$ with $$A \cap C = \{1\}$$. There is no such subgroup. There are subgroups such as $$\{1,r^2,s,r^2\}$$ (in fact there are two such subgroups) that are isomorphic to $$B$$, but they all contain $$r^2$$.

As an exercise, try to think of an example in which $$A = Z(G)$$ and there are complements of $$A$$ in $$G$$.

I am afraid that I do not really know the answer to your second question. It seems to me like an informal comment, and I would advise you not to worry about it.

• I don't think the question suggests that "central extension" is reserved for the cases where there is no complement. Just that it might happen. Apr 14, 2020 at 10:42
• @CaptainLama The quote says "This extreme is called a central extension" which appears to be referring to the situation where the search for a complement proved to be an utter failure. I don't find this informal language very helpful. Perhaps my search for a complement was an utter failure because I am very poor at finding complements? Apr 14, 2020 at 11:15