Recent content by mehtamonica

Thanks, Micromass.

Show that there are two abelian groups of order 108 that have exactly one subgroup of order 3. 108 = 2^ 2 X 3 ^ 3 Using the fundamental theorem of finite abelian groups, we have Possible abelian groups of order 108 can be : Z108, Z4 + Z27, Z2+Z2+Z27, Z4+Z9+Z3, Z2+Z2+Z9+Z3, Z4+Z3+Z3+Z3...

Thanks a lot, Micromass.

As far as the result goes the external direct product of two abelian groups is abelian...but is the internal direct product abelian too ? i mean if subgroups H and K are abelian can we conclude that the IDP is abelian ?

Thanks, Micromass. If G is the internal direct product of its subgroups H and K ,then the possible orders of subgroups H and K can be 2 and 4 or vice a versa. It seems that both H and K are abelian. How can move further from this ?

I have to prove that D4 cannot be the internal direct product of two of its proper subgroups.Please help.

Thank a lot for this guidance, but i still have a doubt and hope that you may help me out this time as well. when f(x)=0 for x>=1 then how do we substitute x=1 in the solution obtained for 0<=x<1? please enlighten me on this

i have to solve the following differential equation : dy/ dx + y = f(x) where f (x) = 2, 0 <= x < 1 and 0 if x >=1 and y (0)=0. please explain how to solve it as it involves a discontinuous function ? I am stuck while computing after computing the integrating factor e^x. Please suggest how...

Thanks a lot. I got it. If we take H as the subgroup consisting of all rotations of Dn, then being a cyclic group, it would also be abelian. Then again, subgroup K of order 2 is abelian. Further, the external direct product H + K is abelian as H and K are abelian. Thanks !

I have to solve the differential equation (y')^2= 4y to verify the general solution curves and singular solution curves. Determine the points (a,b) in the plane for which the initial value problem (y')^2= 4y, y(a)= b has (a) no solution , (b) infinitely many solutions (that are defined for...

Thanks, you are a great teacher. :smile:

ase 2: If order of d = 3, then we need that lcm ( O(a), O(b), O(c))= 2 It can happen in three ways: (a) O(a) = 2, O(b) = 1 or 2, O(c)= 1 or 2. (b) O(b) =2, O(a) = 1 , O(c)= 1 or 2. (c) O(c)=2, O(a) = 1 , O(b)= 1. According to this, Total no. of elements 16 + 8 + 4 +2 = 30. Please suggest...

Case 2: If order of d = 3, then we need that lcm ( O(a), O(b), O(c))= 2 It can happen in three ways: (a) O(a) = 2, O(b) = 1 or 2, O(c)= 1 or 2. (b) O(b) =2, O(a) = 1 or 2, O(c)= 1 or 2. (c) O(c)=2, O(a) = 1 or 2, O(b)= 1 or 2. According to this, in each case, there can be 8 elements in...

I have to find the number of elements in Aut(Z720) with order 6. Please suggest how to go about it. 1) Aut(Z720) isomorphic to U(720) (multiplicative group of units). 2 ) I am using the fundamental theorem of abelian group that a finite abelian group is isomorphic to the direct products of...

The dihedral group Dn of order 2n has a subgroup of rotations of order n and a subgroup of order 2. Explain why Dn cannot be isomorphic to the external direct product of two such groups. Please suggest how to go about it. If H denotes the subgroup of rotations and G denotes the subgroup of...