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In regular everyday English if you say something like "A was extended by B to get C", to me it means that A was in existence, B was added onto it, and now there is a larger object C. For example, "the city was extended by 20 square kilometers."

To me, this is consistent with the mathematical terminology for field extensions. We say things like $\mathbb{C}$ is a field extension of $\mathbb{R}$; The field $\mathbb{R}$ is "surrounded" by $\mathbb{C}$.

But with group extensions, it doesn't work that way, as was pointed out to me here. If $H$ is extended by $K$ to get $G$, it's not $H$ that is "surrounded" by $G$, but rather $K$ that is "surrounded" by $G$.

My questions are:

  1. Is there an alternate understanding of group extensions that make the terminology more consistent with my understanding of everyday English?
  2. Does anyone know the original instance of using the term "extension" for groups the way it is currently used? In particular, I wonder if the idea was first defined in a different language, mistranslated, and stuck.
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2'5 9'2
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2 Answers2

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Hmmmmm... I checked my books; here's the ones that include a definition:

According to Rotman's Introduction to the Theory of Groups, 4th Edition,

Definition. If $K$ and $Q$ are groups, then an extension of $K$ by $Q$ is a group $G$ having a normal subgroup $K_1 \cong K$ with $G/K_1\cong Q$.

Same definition appears in D.J.S. Robinson's A Course in the Theory of Groups, 2nd Edition.

Marshall Hall's The Theory of Groups states at the beginning of Chapter 15 ("Group Extensions and Cohomology of Groups"):

Generally speaking, any group $G$ which contains a given group $U$ as a subgroup is called an extension of $U$. [...] Here, however, we shall consider only cases in which $U$ is normal in $G$.

He does not seem to define extension in general, but he does say later

Let us suppose that all factors $(u,v)$ in an extension of a group $A$ by a group $H$ lie in the center $B$ of $A$. Then we shall say that $E[A,H,a^u,(u,v)]$ is a central extension of $A$ by $H$.

In the notation that Hall has established, $E[A,H,a^u,(u,v)]$ represents a group in which $A$ is normal, and the quotient modulo $A$ is isomorphic to $H$. So this agrees with the nomenclature in Rotman and Robinson (the book was written in 1959).

Scott's Group Theory (published in 1964), defines:

An extension of $H$ by $F$ is an exact sequence $$1\to H\to G\to F\to E.$$

On the other hand, Isaacs' Finite Group Theory has the opposite convention (p 66):

Given groups $N$ and $H$, a group $G$ is said to be an extension of $H$ by $N$ if there exists $N_0\triangleleft G$ such that $N_0\cong N$ and $G/N_0\cong H$.

He does note, however:

As we mentioned, however, this use of prepositions is sometimes reversed in the literature, so readers should attempt to determine the precise meaning from the context.

Bourbaki (Algebra I.6.1) gives:

Definition. Let $F$ and $G$ be two groups. An *extension of $G$ by $F$ * is a triple $\mathscr{E}=(E,i,p)$, where $E$ is a group, $i$ is an injective homomorphism of $F$ into $E$ and $p$ is a surjective homomorphism of $E$ onto $G$ such that $\mathrm{Im}(i)=\mathrm{Ker}(p)$.

(That is, the usage agrees with Isaacs)

This does not really answer your question about the history; but it does show that both usages are common in the literature, and that the one you call "consistent with everyday English" has been around for at least fifty years or more. I would try to take a look at Schreier's original paper on factor sets and extensions to see which use was original, and which one was introduced later...

Added. I thought about it some more; here's a guess:

If you think of groups as sets with a binary operation on it, then it makes more sense to think of an extension $$1\to N \to G\to Q\to 1$$ as "an extension of $N$ by $Q$", because $G$ "extends" $N$ in the sense that it contains (a copy of) $N$, and the operation on $G$ is an extension (in the function sense) of the operation on $G$.

However, groups were not originally thought of this way. Originally, groups were objects that acted on sets. (Burnside calls the elements of a group "operations"). If you have an action of $Q$ on a set $X$, then given an extension $$1\to N\to G\to Q\to 1$$ there is a natural way to extend that action to all of $G$; by contrast, if you have an extension as above and an action of $N$, it may be impossible to "extend it" to an action of $G$. As an example, the natural action of $S_6$ on $\{1,2\ldots,6\}$ cannot be extended to $\mathrm{Aut}(S_6)$, even though we have an extension $$1 \to S_6 \to \mathrm{Aut}(S_6)\to C_2\to 1.$$ Here, if you think of groups as "operations on a set", it makes more sense to think of $G$ as "extending" the set of operations from $Q$ to $G$.

So thinking in terms of actions (as you would if you do a lot of representation theory or character theory, which is what Isaacs does, for instance, or what Ken Brown does in Cohomology of Groups), then the terminology that calls $G$ an extension of the quotient by the kernel makes more sense. Thinking in terms of sets with an operation on it makes it so that it makes more sense to call $G$ an extension of the kernel by the quotient.

Again, this is just an (informed) guess; looking at Schreier's paper would likely settle which one was first or why.

t.b.
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Arturo Magidin
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  • Thanks for all of these references. The older references are consistent with how I would guess the terminology should be, and it's nice to see Isaac's note about mixed prepositions. Modern references I've looked at (like Wikipedia) should have similar notes. I imagine the Bourbaki text was originally in French; I still wonder if a mistranslation is the historical explanation for all of this. – 2'5 9'2 Sep 14 '11 at 22:44
  • @alex: I don't think it's a mistranslation. I'm about to add what is just a guess, but which I think is probably close to the mark. Checking Schreier's paper would probably settle it. – Arturo Magidin Sep 15 '11 at 00:08
  • @alex.jordan, Arturo: Schreier's first paper on group extensions is here. In the very first sentence (and later on) he speaks of Erweiterungen von $\mathfrak{A}$ mit Hilfe von $\mathfrak{B}$ "extensions of $\mathfrak{A}$ with the help of $\mathfrak{B}$" and he means that $\mathfrak{A}$ is the normal subgroup and $\mathfrak{B}$ is the quotient. Groups are algebraic structures, not acting elsewhere. Side note: The idea of classifying extensions is attributed to Hölder's work on groups of order $p^3, p^2q, pqr, p^4$ in a footnote on the very first page. – t.b. Sep 15 '11 at 00:49
  • @Alex: Dear Arturo and Alex, My experience (as someone who uses homological algebra frequently, but who is not a pure algebraist but rather a number theorist who also discusses frequently with algebraic geometers and representation theorists) is that the terminology of Bourbaki and Isaacs is what is used. (I wasn't aware that the alternative had any legitimacy at all before reading this answer.) The reason is that, in the analogous situation for modules, where $A$ and $B$ are $R$-modules for a ring $R$, the group of extensions of $A$ by $B$ (in the Bourbaki sense), usually denoted ... – Matt E Sep 15 '11 at 01:37
  • ... $Ext^1_R(A,B)$, is computed as a derived functor of the the Hom functor $Hom_R(A,B)$. For this to be true it is important that extensions of $A$ by $B$ is understood in the Bourbaki/Isaacs sense, not the reversed sense. Regards, – Matt E Sep 15 '11 at 01:38
  • @Matt: But why couldn't $Ext_R^1(A,B)$ read out loud as "extensions by $A$ of $B$", having the same mathematical meaning (derived functor of the the Hom functor), but keep the English connotation that $B$ is within? – 2'5 9'2 Sep 15 '11 at 05:27
  • @Arturo,Theo: Theo's citation could lead to the kind of language blip I'm imagining might have happened. I would translate Schreier just as Theo has, and obtain the Rotman/Robinson/Hall/Scott language. But given the double presence of "von", I could imagine a mistranslation. – 2'5 9'2 Sep 15 '11 at 05:35
  • @Alex: Dear Alex, I'm unsure of what you're asking in your last comment. Because of the ordering of $A$ and $B$, it is natural to read $Ext^1(A,B)$ as "extensions of $A$ by $B$". On the other hand, if this is to be read as the derived functor of $Hom(A,B)$, we don't have any choice as to its precise interpretation (because $Hom(A,B)$ is not symmetric in $A$ and $B$), and it turns out (when one computes it) to correspond to extensions in which $A$ is the quotient and $B$ is the subobject. Does this address your question at all? Best wishes, – Matt E Sep 15 '11 at 12:48
  • @Theo: Thanks for the link to Schreier. (For what it's worth, which is little, I use the nomenclature or Rotman and, apparently, Schreier. I know that groups are "algebraic structures, not acting elsewhere", but originally they were sets of operations. Burnside's book defines groups by saying "Let $A,B,C,\ldots$ represent a set of operations which can be performed on the same object or set of objects. Suppose this set of operations has the following characteristis. ($\alpha$) The operations of the set are all distinct so that no two of them produce the same change (cont) – Arturo Magidin Sep 15 '11 at 13:25
  • @Theo: (cont) in every possible application. ($\beta$) The result of performing successively any number of operations of the set, say $A, B,\ldots, K$ is another definite operation of the set, which depends only on the component operations and the sequence in which they are carried out, and not on the way in which they may be regarded as associated. Thus, $A$ followed by $B$ and $B$ followed by $C$ are operations of the set, say $D$ and $E$; and $D$ followed by $C$ is the same operation as $A$ followed by $E$. (cont) – Arturo Magidin Sep 15 '11 at 13:26
  • @Theo (cont) ($\gamma$) $A$ being any operation of the set, there is always another operation $A_{-1}$ belonging to the set such that $A$ followed by $A_{-1}$ produces no change in any object. The operation $A_{-1}$ is called the inverse of $A$. The set of operations is then said to form a Group." So Burnside thinks of the groups as necessarily acting on something, and this was by no means an uncommon or idiosyncratic view at the time. – Arturo Magidin Sep 15 '11 at 13:28
  • Sorry, the telegram style distorted my intention and I wasn't trying to educate or correct you -- on the contrary. That fragment of a sentence was meant to say: "[In Schreier's text] groups are [viewed as] algebraic structures" -- as opposed to 'transformation groups' -- in order to support your point that in this interpretation the nomenclature used by Schreier and Rotman and you is the (more) natural one. Thanks for that excerpt of Burnside. I completely agree with all the things you're saying in your comments. – t.b. Sep 15 '11 at 13:48
  • @Theo: No worries; glad that it is cleared up. – Arturo Magidin Sep 15 '11 at 16:04
  • @Alex: In summary, I think that it is not a case of mistranslation, but rather a case of getting "at" the construction from different points of view. If you are coming from a point of view of actions, of homology, of cohomology, of derived functions, then it makes more sense to talk about "extension of the quotient by the normal subgroup". If you are coming from the point of view of a set with an operation, then it makes more sense to talk about "extension of the normal subgroup by the quotient." Curiously, the only sources where I found the warning about the two uses are the former ones/ – Arturo Magidin Sep 15 '11 at 16:06
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    @Matt: I had the same question as Alex about your comment. I believe the point is that there is no disagreement about what $Hom(A,B)$ should mean (morphisms from A to B), so if you want $Ext^1(A,B)$ to mean "extensions of $A$ by $B$" and you want to preserve the relation that $Ext^1(A,B)$ is a derived functor of $Hom(A,B)$ then there is only one way to define "extensions of $A$ by $B$". – Ted Sep 16 '11 at 07:09
  • @Ted: Dear Ted, Yes, this is my point (such as it is). Regards, – Matt E Sep 16 '11 at 12:31
  • @Arturo: thanks for the action viewpoint. It's too bad people don't call it "inflation of Q by N" like they do in the specific case of representations. – Jack Schmidt Sep 25 '11 at 16:07
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I have come to my own answer to my first question. (It looks like the unusually long comment thread on Arturo's answer leads to the same basic place - see Arturo's comment currently at the end of the thread. If anyone feels like upvoting this answer, consider upvoting a comment in that thread instead/too.)

If the quotient of two groups has an action on some object, then the parent group can act on that object as well. I find this consistent with the English of extending a small thing to a larger thing. It's a little more like extending a group action to a larger group action, rather than a group to a larger group, but tomatoes, potatoes, tomatoes, potatoes.

And this answer to the first question makes the second question less worthy of consideration. Although I want to thank several people involved in Arturo's answer for documenting how both language conventions are in use.

2'5 9'2
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