Generalized quaternion group

Define the generalized quaternion group by the presentation Q_n = < a,b|a^2n=1, b^2=a^n, ab=ba^-1 > for n >=1 Show that |Q_n|=4n Show that Q_1 is cyclic and Q_2 is the quaternion group Show that Q_3 is not isomorphic with either the dihedral group D_6 or the alternating group A_4

Group with presentation

Let G = < a,b|a^4=b^3=1,ab=ba^3 > Show that G is cyclic of order 6.

Group proof

Consider the cartesian product H=Z_2 X Z_n (as a set).Define a binary operation on H to be (i,j)*(k,l)=(i+k,(-1)^k *j+l), where i,j,k and l are vectors. Show that H is a group under this operation, determine its order. Let G = < a,b|a^n=1, b^2=1, abab^-1=1 > Show that |G|<=2n. Show that H is isomorphic to G. ...continues

Semidirect product or split extention

If H and K are subgroup of G, with K a normal subgroup of G, K intersect H=1 and KH=G, then G is called a semidirect product (or split extention) of K by H. If sigma=(12) in S_n, n>=2, show that s_n is a semidirect product of A_n by Show that the dihedral group D_n= is a semidirect produc ...continues

Abelian Group

Let G be a group in which a^2 = e for all a in G. Prove that G is an abelian group. Thanks.

Homomorphisms

Please attached for the question to help me understand Group and Ring Homomorphisms. Is this function a Homomorphism? a) Does 1:22 :22 where 1(a1,a2) = (a1,0) define a group homomorphism? a ring homomorphism? Prove your answers. b) Does 1:22 :22 ...continues

Understanding Kernel

Please help me to understand the attached homework problem on finding the kernel. Find the Kernel Consider the groups of nonzero complex numbers, *, and positive real number +, both with multiplication. Let f: *  + be defined by f(z)= |z|. Prove that f is a homomorphism and find ...continues

Set base Ring

Could you please help me to understand Rings and its Ideal. Please see the attached practice problem. Thanks The symmetric difference operation on sets is , defined by X  Y = (XY)-(XY). Let A be any set and let R=(A) be the set of all subsets of A — the "power set" of A. We wi ...continues

Group Theory Questions

I need step-by-step solutions to the following group theory questions. The price is set to 13 credits but of course if you're interested and willing to do this for more credits, please let me know how much you want for solutions in Microsoft Word/Equation Editor or MathType and how much time will you need. I'll pick the best ...continues

Statisticsanalysis of variance tests and quality of group means

I Really need help with this and do not undertstand it very well. I would appreciate anyone's help.