graphicalUSSRinterface posted:insta_gramsci can you explain cosets, conjugates and normal subgroups to me?
Sure! Cosets first:
So if G is a group, H is a subgroup of G, and g is any element (member) of G, then the set gH = { gh | h is in H } is a left coset of H in G. (The formation of right cosets is analogous). Intuitively it's the set of all "multiples" of elements of H by the group element g. For different choices of group elements g you may or may not get different cosets--for example, if g = h for some element h in h, then gH = H (the same if g is set to be the identity element of G).
Are you familiar with equivalence relations? They're essentially a generalization of the notion of equality. For an arbitrary set S with elements x,y, and z, an equivalence relation, denoted by ~ (the tilde can read verbatim as "is related to") is some conditional statement such the following conditions hold:
1.) Symmetry: x ~ x for all x in S
2.) Reflexivity: If x ~ y, then y ~ x for all x, y in S
3.) Transitivity: If x ~ y and y ~ z, then x ~ z for all x, y, z in S
We can define a relation ~ in G by "x ~ y if and only if there exists h in H such that x = yh". In other words, two elements x and y are "equal" provided that they lie in the same coset of H. Let's make sure it's an equivalence relation:
We first assume that x is some "h-multiple" of y--i.e. there exists h in H such that x = yh.
1.) Since any subgroup contains the identity 1, setting h = 1 gives x = x(1) = x, or x ~ x;
2.) Since x = yh, and subgroups are closed under taking inverses, it follows that y = xh^{-1}, so that y ~ x ;
3.) This one is especially boring to type but it follows from 1.) and 2.).
So our relation ~ is an equivalence relation of G--we're saying that we can think of two group elements x and y as "equal", or as belonging to the same coset, provided that one is an h-multiple of the other.
So the cosets of H in G actually serve as a partition of G, in the same way that you can draw a circle and split it by drawing lines through it. I'll connect this to conjugation next and then to normal subgroups after.
Now, within the group G, two elements x and y are said to be conjugate to each other if there exists g in G such that gxg^{-1} = y.
Now, for an abelian group, xy = yx holds for any two elements x and y in G, which (by inversion) is equivalent to xyx^{-1} = y, so in an abelian group any two elements are conjugate.
So conjugating elements, as an "operation", is a generalization of commutativity. Abelian groups are very friendly to work with since you can take multiplication for granted--distinctions between left- and right- whatever are unnecessary. In other words multiplying elements doesn't really give you a lot of useful information about the group's structure. But for non-Abelian groups, conjugation is a useful tool to further study the group and it's the "closest we can get" to commutativity in a practical (or at least computational) sense.
That's conjugation as an "operation"--let's connect it to cosets and then to normal subgroups:
A coset is the set of all multiples of a subgroup H in G by a fixed group element, say g. So for some arbitrary h in H and g in G, the group element gh is a member of the left coset gH. Analogously for right cosets, hg would be a member of Hg.
Suppose that G is an abelian group. Then gh = hg, and since this holds for any h in H, it follows that gH = Hg. So in an abelian group the left and right cosets coincide, i.e. they give no new information about the group.
For a non-abelian group, however, it's not necessarily true that gH = Hg. As sets of objects gH and Hg could be completely distinct. But when they do coincide for every choice of g in G, we recover a certain amount of "abelian-ness" within a non-abelian group. Such subgroups, whose left and right cosets coincide, are said to be normal subgroups of G--
A subgroup H of G is said to be normal in G if gH = Hg for every g in G.
This definition is equivalent to this one, which is formulated more in the "conjugation as an 'operation'" sense from earlier:
A subgroup H of G is said to be normal in G if for every h in H and for every g in G, ghg^{-1} is in H.
Now, from here, normal subgroups become an extremely important tool in the classification of groups and also in the formation of new groups. But intuitively, or at least in a motivational sense, you can think of them as being an attempt to recover as much "abelian" behavior in a non-abelian group as possible, which might be one of the reasons they are called "normal".
And any axiomatic system that purports to be able to demonstrate its own validity is actually
invalid
looks like he'll sink it in one, but if it rolls wide he can still probably eagle it
ialdabaoth posted:if im reading this graph right, "stabilized earth" is markedly less "stable" than "hellhouse earth"
What that means is that a huge amount of energy would be required to move the earth from the hothouse earth state. "Stability" doesn't mean stability of modern society or anything like that, it means that more energy is required to move Earth out of the Hothouse state than it takes to move Earth now, in the less stable state.
So, let's say we're putting in X trillion BTU equivalents today to warming the Earth. It's relatively more easy to change its state today than in Hothouse Earth. Assuming that society sticks around, we might need 3*X (for example) trillion BTU equivalents to get Earth back to where it is today, once we've reached Hothouse Earth.
ialdabaoth posted:if im reading this graph right, "stabilized earth" is markedly less "stable" than "hellhouse earth"
theres literally an axis labeled "stability"
toyotathon posted:i'm trying to determine why degrees of freedom in kinematics are in the set of natural numbers, and the answer at first blush is, because spacial dimensions are in the set of natural numbers. but then why is that so
this is true but to ask "why?" can be misleading. a more practical question might be "what would it mean for a 'space' to have a dimension not in the natural numbers?" there are a few different ways of approaching this, fractals can be said to have non-integer dimensions which has to do with their self-similarity and scaling under transformations (effectively a fractal embedded in some number of dimensions can have fewer "effective" dimensions arising from its self-similarity). there are also spaces with an continuously infinite number of dimensions, partial differential equations have solutions in spaces like these but each of these dimensions corresponds to a specific point in physical space so it might not be what you have in mind. generally, in mathematics, you can always generalize the concepts you're working with in any number of ways, the trick is to find generalizations that are productive and useful.
The short version of that i guess is that i think math is probably harder to self study so getting into it in a structured environment like a university course is great. i was incredibly average at math until college where i had to sit and do it over and over again.
Edited by c_man ()
Caesura109 posted:You cant imagine how intimidating this thread is to me
i volunteer at the library tutoring a woman for her ged. we recently got to the math part and when we made it to algebra - basic, 2x + 3 = 7 type stuff - and i was fumbling through it like an idiot. i never learned calculus so i will be even more useless going forward. so yes, i can imagine
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taxation is theft vote Ron Poal