Recent content by zcdfhn

Prove or disprove: Suppose N is a normal subgp of G and N' is a normal subgp of G'. If G is isomorphic to G' and N is isomorphic to N' does that mean that G/N is isomorphic to G'/N'? I was trying to work out a proof until my professor told us to think of subgroups of the integers when doing...

Thank you so much! You really clarified my view on quotient sets, I finally found a connection to what I learned about the quotient Z/nZ and it makes so much sense now.

Prove R/Q has no element of finite order other than the identity. First of all, I have trouble visualizing what R/Q is. But I do know that afterwards you can try to raise an element in R/Q to a power to get to 0, but there will not be a finite number that will be able to do so except zero...

Thanks that's much clearer, so my approach to find all the quotient groups of D6 is to use the 1st isomorphism theorem, so I start with finding all the normal subgroups of D6, which are {id}, {id, R^2, R^4}, {id, R, R^2, R^3, R^4, R^5}, and D3, where R = rotation by pi/3 (I'm not sure if I'm...

Find, up to isomorphism, all possible quotient groups of D6 and D9, the dihedral group of 12 and 18 elements. First of all, I don't understand the question by what they mean about "up to isomorphism." Does this mean by using the First Isomorphism Theorem? Also does this question imply that...

Oh ok i think I get it, so rational numbers that are the same signed fractional distance from an integer are in the same coset of Q/Z?

I was wondering what Q/Z, or the rational numbers modulo the integers. I am struggling to visualize what the cosets may be. Thank you for your time.

I think I had a moment of insight when I was learning about isomorphisms and I just need people to verify what I think I figured out. So I was thinking if two sets are isomorphic to each other does that mean ANY bijective function from one set to the other is an isomorphism? Thank you for...

A uniform solid sphere of mass M and radius R rolls, without slipping, down a ramp that makes an angle θ with the horizontal. The question ask for me to find the force of friction between the ramp and the sphere. My attempt at the problem was to utilize the x-component of the force of gravity...

A glide reflection composed with itself is a translation. And you get a glide reflection by composing a reflection with a translation.

Let G1 be the group generated by a nonzero translation and G2 be the group generated by a glide reflection. Show that G1 and G2 are isomorphic. Here is how I started: G1 = <Tb> where b\in C and Tb(z) = z+b G2 = <ML \circ Tc> where c and L are parallel to each other. Let's define a function...

Suppose g\in Isom C, z\in C: Prove that the g-orbit of z is invariant under g. I just need some clarification on what this is asking for: 1.) Are we assuming that g is a group of the isometries of C under composition? 2.) To show invariance, would I only have to show that the g-orbit...

I know M(z) = a(z-bar)+b, where |a| = 1 and a \neq 1, a(b-bar) + b = 0 and i also know M(z) = z0 + ei2\eta(zbar - z0bar)

Prove that every non-zero translation Tb is a composition of two reflections in lines which are perpendicular to the direction of vector b. Note: Tb(z) = z+b where z,b\inC My guess at how to start this is to assume b = rei\theta where r = |b| and \theta=arg b, so then the direction of b is...

I have to prove that for all k,m,n \in N that if m+k = n+k, then m=n. The problem mentions that I must prove this by induction. I did the base case k = 0: If m+0 = n+0, by identity m=n. Then I attempt to show that m+1 = n+1 implies m=n, but I am stuck, I don't see how induction can be...