In my text, I often come across words like "there exists a canonical homomorphism between the group and the quotient group " for example -in the first isomorphism theorem . I do not understand what is meant by the word "canonical" in such contexts although I sort of get what is going on. May I request a clarification of the concept here? Thanks.
"canonical" in this instance means "standard" (and in this instance is actually defined by a universal property).
meaning: we do it in "the same way" no matter which particular group and normal subgroup we have.
ok, let's look at it this way:
suppose i have a group G, and a homomorphism f from G to some other group G'. suppose that H is a subgroup of G, and i say "f kills H". what might i mean by that?
what i mean is: we have an obvious homomorphism of H into G defined by:
iH(h) = h, for all h in H. and when i say f kills H, what i mean is:
f∘i is the 0-homomorphism (that is: it maps everything to the identity e' of G'...the term "0" is somewhat mis-leading, we mean its a "trivial" homomorphism).
now if N is a normal subgroup of G, we have a surjective homomorphism p:G-->G/N given by p(g) = gN (send every element of G to the coset of N it lives in).
p is said to be "universal among homomorphisms that kill N". well, what does THAT mean?
it means if we have a homomorphism f:G-->G', such that f(n) = e' for all n in N, f "factors through p", that is there is some OTHER homomophism f' with f = f'∘p.
so let's unravel this. if f kills N, what we are saying is N is contained in ker(f), let's call ker(f), K (for kernel).
what is this other homomorphism f'? it seems like it should be:
f'(gN) = f(g), right? does this even make sense?
well, first we need to check "well-defined-ness". that is we need to be sure that if gN = hN, that f(g) = f(h).
if gN = hN, then h-1g is in N. but N is contained in K, so h-1g is also in K, which means f(h-1g) = e' (since K is the kernel of f).
since f is a homomorphism, f(h-1g) = f(h-1)f(g) = [f(h)]-1f(g). since this also equals e', we have:
f(g) = f(h) (multiply by f(h) on the left). so f' is indeed at least a function from G/N to G'.
to see that f' is actually a homomorphism, we compute:
f'((gN)(hN)) = f'((gh)N) = f(gh) = f(g)f(h) = f'(gN)f'(hN).
finally we check that f'∘p = f:
f'∘p(g) = f'(p(g)) = f'(gN) = f(g).
this is all pretty much the same thing you see in the first isomorphism theorem, it's not all THAT exciting. so why bother?
well...because proving things about N as a subgroup of G, and about G/N as a factor group (quotient group) usually depends on (like we did above) picking "typical elements" g,h in G and seeing what happens to them.
but saying: p is universal among homomorphisms that kill N isn't a statement about ELEMENTS of G, its a statement about our "canonical" homomorphism p. in other words, instead of looking at the OBJECT G/N, we're looking at the MAPPING p.
and we're saying: "if a homomorphism (any homomorphism) kills N, it has to "go through p" first. p is "special" or "distinguished" among all the homomorphisms that kill N, it's OPTIMAL (it kills N, and ONLY N, not anything more).
that is: given the problem of a group G, and a normal subgroup N, how do we make a homomorphism that "just kills N"? form the factor group (apply the homomorphism p) G/N.
this construction occurs so frequently, we don't even bother giving p "a name", because we do it "the same way" for every group G and any normal subgroup N.
in this light, what the first isomorphism theroem says is:
there is no practical difference between a factor group G/N and a homomorphic image of G. we can use either "construction" as it suits us. sometimes the quotient group G/N is easier to understand, sometimes f(G) is easier to deal with.
so rather than carry the letter p around and explicitly define it (for a particular subgroup N, and a particular group G), we just say:
consider the canonical homomorphism G-->G/N, which you are expected to know means "p".
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