Understanding Homomorphisms: Onto vs Into in Group Theory

[swartzism]

"Onto" vs "into"

What is the difference between a homomorphism which is from a group G ONTO a group H and a homomorphism which is from a group G INTO a group H?

 

[radou]

"Into H" only means that the codomain of the mapping is H, and "onto H" means that the codomain equals the image of the mapping, i.e. it is surjective.

 

[Hurkyl]

Sometimes, into is used to imply injective.

 

[Landau]

Sometimes, into is used to imply injective.

Can you give an example / reference? I have never come across this meaning.

 

[ABarrios]

Can you give an example / reference? I have never come across this meaning.

Same, all the books I have seen in Algebra use onto when the function is surjective, otherwise they say into.

 

[Fredrik]

It should also be mentioned that "into" doesn't imply that the function isn't surjective. We can definitely talk about a surjection from X into Y. "Into" is the word you use by default, and you can change it to "onto" if you're allergic to French or something*, so that you need to say that the function is surjective without actually using that word.

*) I don't know if "surjective" should be considered a French word, but I read that the term was introduced by Nicolas Bourbaki.

 

[Tac-Tics]

"Into" is a meaningless word.

"Onto", on the other hand, is a synonym for "surjective". (Sur = on in French, that's how I remember it). A surjection is a functions where the range equals the codomain.

Surjections are usually studied together with injections (often called "one-to-one" functions). A function is surjective and injective, we call it bijective (or "one-to-one onto").

 

[Martin Rattigan]

"Into" is a meaningless word.

Wot?

 

[D H]

Sometimes, into is used to imply injective.

That is exactly that I was taught, years and years (decades and decades) ago. For those of us who were allergic to French, into=injective, onto=surjective, into and onto=bijective.

 

[g_edgar]

I think you will get into trouble nowadays if you assume "into" means injective.

 

[Martin Rattigan]

That is exactly that I was taught, years and years (decades and decades) ago. For those of us who were allergic to French, into=injective, onto=surjective, into and onto=bijective.

I bet my father's older than your father, but neither into nor onto meant 1-1 when I was at school!

 

[lavinia]

into means injective - when you say a mapping is into. It means nothing if you say a map takes a space into another.

 

[Rasalhague]

My very limited experience matches what Radou and Fredrik have said, e.g.

Carol Whitehead: Guide to Abstract Algebra:

A relation $\alpha$ which is a mapping between a set X and a set Y is called simply a mapping of the set X into the set Y. We write:

$$\alpha: X \to Y$$

The departure set, X, is called the domain of the mapping and the 'arrival' set, Y, is called the codomain.

It's only later that she introduces the concept of injective (one-to-one) mappings.

On the other hand, Borowski & Borwein: Collins Dictionary of Mathematics:

3. (also as adj.) (of a function) having an IMAGE contained within (a given set). For example the function y=x² maps the integers into the nonnegative integers; in some usages a ONE-TO-ONE mapping is both into and ONTO. See INJECTIVE.

On it's own, given their own definition of "into", the final comment is confusing to me. But the entry INJECTIVE says that, according to one usage, one-to-one means injective (this is the usage I'm familiar with), whereas, according to another usage, one-to-one means bijective. So maybe that final comment is talking about the one-to-one = bijective usage, in which case maybe they're saying that, according to that usage, into means injective.

I reckon it's a consipracy to keep us all using French, allergic or not ;-)

 

[Martin Rattigan]

"Into" is a meaningless word.

It means nothing if you say a map takes a space into another.

I can't understand why people keep saying "into" is meaningless. Why doesn't it mean into? What exactly is supposed to make it meaningless compared with the other terms describing mappings?

I'm mystified. Can someone give me a clue.

 

[lavinia]

I can't understand why people keep saying "into" is meaningless. Why doesn't it mean into? What exactly is supposed to make it meaningless compared with the other terms describing mappings?

I'm mystified. Can someone give me a clue.

into is used in two ways.

- A maps takes one space into another. Here it tells you nothing about the map.

- A map is into. Here into means that the map is injective.

 

[Martin Rattigan]

into is used in two ways.

- A maps takes one space into another. Here it tells you nothing about the map.

- A map is into. Here into means that the map is injective.

So you presumably mean it's ambiguous to the point of being not worth using?

Doesn't the first statement imply that none of the images of the map fall outside the second space? This is the usage I am used to. The books I have that use the English descriptions seem to consistently use it to mean exactly that, though they also tend to use "to" and "into" interchangeably.

By "injective" I would understand 1-1. Is that also your understanding? I am another one who has never seen "into" used in this way. Where is this use from?

 

[D H]

So you presumably mean it's ambiguous to the point of being not worth using?

No, it appears to me that lavinia is saying the word has multiple meanings. That does not mean its ambiguous. The meaning is usually rather clear from context and usage.

For a good example of a word (or phrase) that is closer being too ambiguous is 1-1. You are interpreting that as meaning injective. Others use it to mean bijective.

 

[lavinia]

So you presumably mean it's ambiguous to the point of being not worth using?

Doesn't the first statement imply that none of the images of the map fall outside the second space? This is the usage I am used to. The books I have that use the English descriptions seem to consistently use it to mean exactly that, though they also tend to use "to" and "into" interchangeably.

By "injective" I would understand 1-1. Is that also your understanding? I am another one who has never seen "into" used in this way. Where is this use from?

The use of into as injective is standard. You should expect to see it used that way.

 

[Landau]

Please, give references for 'into=injective' being standard. I don't recall having read a book which uses this terminology.

 

[D H]

Stop quibbling over semantics. Several members have noted that they have either been taught or know that others have been taught that into can be a synonym for injective.

into is used in two ways.
- A maps takes one space into another. Here it tells you nothing about the map.
- A map is into. Here into means that the map is injective.

In the first meaning into is used as a preposition. In the second it is used as an adverb. There is *no* ambiguity here.

One-to-one on the other hand is ambiguous. To some authors one-to-one is a synonym for injective while for others it is a synonym for bijective. A reader who comes across that term in a text or a paper had either know what that particular author means by that term or infer the meaning from its usage.

Just because a word has multiple meanings does not mean that the word is useless. Here is a graph:

Here is another graph:

Does these multiple meanings mean we should avoid using the word "graph"? Of course not.

 

[Martin Rattigan]

...
For a good example of a word (or phrase) that is closer being too ambiguous is 1-1. You are interpreting that as meaning injective. Others use it to mean bijective.

I actually use it to mean both.

I will say e.g. f is a 1-1 mapping from A onto B or f is a 1-1 correspondence between A and B to mean f is a bijection from A to B, or f is a 1-1 mapping from A into (or just to) B to mean f is an injection from A to B.

I think the use of the words "into", "onto" or "between" in conjunction with the phrase "1-1" should be enough to distinguish the uses.

 

[Martin Rattigan]

No, it appears to me that lavinia is saying the word has multiple meanings. That does not mean its ambiguous. The meaning is usually rather clear from context and usage.

The original question was why people regard "into" as meaningless. I'm genuinely perplexed by this. From lavinia's reply, I understood that ambiguity was the problem. If not then presumably it is to do with, "Here it tells you nothing about the map". Unfortunately, I don't understand that any better, it seems to mean, in that context, what I always thought it meant.

 

[Martin Rattigan]

The use of into as injective is standard. You should expect to see it used that way.

OK, but you didn't actually answer my question. By "injective" I would understand 1-1. Is that also what you mean by the term?

I would like to be sure it's the term "into" that I need to reappraise rather than "injective". (I have seen descriptions of maps as "into" meaning "not onto", but this is not what I'd currently understand by "injective".)

 

[D H]

The original question was why people regard "into" as meaningless. I'm genuinely perplexed by this.

The word is more or less meaningless when used in the sense of "f maps X into Y." Here, the word to could be substituted for into with no change in meaning. In this context, into is a preposition, and just an idiomatic preposition at that. Think of "f maps X into Y[/i] as an active voice way of saying "Y is the image of X under the mapping f" (yech).

From lavinia's reply, I understood that ambiguity was the problem.

I think you misread lavinia's reply. The second meaning as a synonym for injective is quite specific and meaningful. So the word is used in two distinct ways. So what? Lots of words/phrases have multiple meanings, including the jargon used by mathematicians. (Jargon is yet another term with multiple meanings, some of them a bit disparaging. Here I am using the word jargon in the non-disparaging sense of "the technical terminology or characteristic idiom of a special activity or group". )

 

[Martin Rattigan]

OK I'll assume "meaningless" meant synonymous with "to" (though that's another word that seems to have changed its meaning since I was alive).

"Y is the image of X under the mapping f", would normally convey to me, "f maps X onto Y". I'm hoping that was just a slip.

It's the second meaning that worries me.

Given that there are three people who have posted to the forum saying "into" can imply "injective", the meaning of either "into" or "injective" has changed since I was at university in the 60s. I think I've seen, "f is an into map", used to mean, f is not onto some particular set, so it crossed by mind that people could be using "injective" to denote what I'd call "strictly into", i.e. "into" but not "onto". I have always used "injective" to mean "1-1 into" (which would also allow the possibility of "1-1 onto"), but, as I said, the meaning of either "injective" or "into" has changed while I wasn't looking.

What I need to know then is what people mean these days by "injective"; that is should I now allow for the possibility that "into" may be used to mean what I'd have previously referred to as "strictly into" and "injective" may be used to mean the same thing (without necessarily indicating 1-1) or does "injective" still mean what I always thought it meant and I just have to allow for the possibility of "into" meaning what I always thought "injective" meant.

Hence my repeated queries (so far unanswered) about whether people still understand "injective" to mean "1-1". Anybody?

 

[D H]

Regarding the meaning of injective, etc.: Suppose f is a mapping $f : X \rightarrow Y$. The mapping is

• Injective if for every point $y\in Y$ there exists at most one point $x\in X$ that maps to this y.
• Surjective if for every point $y\in Y$ there exists at least one point $x\in X$ that maps to this y.
• Bijective if it is both injective and surjective; i.e., if for every point $y\in Y$ there exists exactly one point $x\in X$ that maps to this y.

Example: The mapping $f : \mathbb R \rightarrow \mathbb R : x\in \mathbb R \mapsto x^2$ is neither injective nor surjective. Restricting the domain to the nonnegative reals makes the mapping injective but not surjective. Restricting the codomain to the nonnegative reals makes the mapping surjective but not injective. Restricting both the domain and the codomain to the nonnegative reals makes the mapping injective and surjective (i.e., bijective).

Hence my repeated queries (so far unanswered) about whether people still understand "injective" to mean "1-1". Anybody?

I've answered this multiple times. One-to-one is a synonym for injective to some people but is a synonym for bijective to others. You had dang well better figure out what the author means when you run across this term.

 

[Martin Rattigan]

Thanks. Your definitions of the "French" variants remain as I always understood them, so at least its only the term "into" that has become ambiguous.

Sorry for not noticing that you had already implied that "injective" does involve 1-1.

I don't think there is any ambiguity in the term 1-1 by the way. I don't think it's a synonym for either "injection" or "bijection" but relates to the function without regard to the codomain. That is, if a function is taken as a one-many relation, then a 1-1 function is a function that is also a many-one relation. If a 1-1 function is onto, it's a bijection, whereas if it's into (in either sense) its an injection. But that doesn't make the phrase 1-1 ambiguous.

 

[Werg22]

This is why I've come to dislike this forum. A question like this doesn't need so many answers.

 

[swartzism]

Yeah, my question was answered from the first post.