What is the concept of injection and surjection?

[SqrachMasda]

Show that a 3 cycle(a,b,c) in Sn has order 3.
Show that a product of disjoint transpositions (a,b)(c,d) in Sn has order 2.
Find the order of (a,b)(x,y,z) in Sn, if this is a product of disjoint cycles.
Apply this to tell the orders of all permutations in A4.

 

[matt grime]

this answers the questions I asked in the other thread then.

you need to learn what the notation means.

reread your lecture notes. it's surprising how often that answers a lot of questions. if you don't have lecture notes, reread the chapter in the book.

your issue seems to be not that you can't do groups but that you don't know what the symbols mean, is that a fair assessment.
For instance, if I were to say, well, why don't you just SHOW (abc) raised to the third power is the identity, since it is just a calculation, where would you have difficulty? the notation? the definition of order?

 

[666yguy]

Title ?

The Achille's heel of most technical communications on the web is the lack of clear communication. It seems a rule that many people with advanced knowledge treat everyone else as being conversant with virtually every facet of their technical field except the few tiny details which they want to discuss. This leaves the less obsessed out of the discussion completely, even though they may have a general interest in the subject. In the present case, I suggest that no one will think less of you for backing up and devoting a sentence each to explaining those technical aspects preliminary to your question, such as the meaning of jargon or notations. You may bring in a surprising number of participants who would otherwise be excluded. The usual response to my making a suggestion like this is implacable hostility from "nerds". I am hoping that in this extraordinary science forum that Dr. Kaku is experimenting with, this simple idea will be well received and even implemented. Keep in mind Dr. Kaku's example, as he explains in simple terms that which he could easily use to snow his audience.

 

[SqrachMasda]

good call 666
yeah, I'm just trying to learn this stuff, but some questions are not clear.
i am not in a class
the book is too abstract, no examples
but it is true the only part of mathematics i have always had trouble with is anyhing dealing with groups, 1-to-1, onto or proofs in general...whenever this stuff comes into math i am lost. I can't grasp the concepts and i don't know why.
i am probably a lot better off figuring it out on my own, i was just hoping for simialr examples.

 

[matt grime]

If you explained all that in the beginning that'd make life much easier. For example - show that (abc) has order 3. Well, if we don't konw why you can't do this question we can't help.

Are you happy with linear algebra? All (finite) groups can be realized as (sub)groups of matrices. If you can think geometrically then it may make your life easier.

Let's take S_3 the permutations of the elements. The triangle has three corners. A permutation of three objects can act as a permutation of the three vertices - this means it gives a symmetry of the triangle. Conversely every symmetry of the triangle (rotation, reflection) permutes the vertices, and gives an element in S_3. So S_3 is the same as the group of symmetries of a triangle.

Let's take the notation (abc)

It means a goes to b, b moves to c, and c moves to a (wraps around to the start)

Label the triangle's vertices a,b,c - what symmetry does this correspond to? Rotating the triangle 1/3 of the way round - so doing it three times gets you back to the start - that's what it means to have order 3.

Would you like more general explanations of what order means, how to compose permutations?

 

[matt grime]

Would you even like the concept of injection and surjection explained?

Here's how to think of an injection:

Imagine a function between two sets as a diagram. Divide a sheet of paper into halves. On the left imagine drawing a point for each element in the input set (domain). and on the right a point for each element in the outpt set (codomain). A function is then an assigment of an arrow exactly one starting at each element in the domain and with its head pointing to a unique point on the right hand side.

It is an injection if no two arrows point at the same object.

It is a surjection if all points on the right have an arrow going to them.