Recent content by charlamov

How would you prove that < x_1,x_2 ... | [ x_i , x_j ] =1, i,j \in N , x_1 ^ p = 1, x_{i+1} ^p = x_i , i \in N > is presentation of Z_{p^ \infinity}

thanks, i finally have found that it is not true generally

Is it true that if there is monomorphism from group A to group B and monomorphism from group B to group A than A and B are isomorphic? i need some explanation. thx

thanks, but i need some upper estimation of number of non-isomorphic groups of order 24 with some explanation.

how would you estimate number of groups of order 24? i do not need exact number, it is 15. I know that there are 5 groups of order 8 and there is 1 , 4 or maybe 7 sylow 3-groups. but i do not know what to do next

i do not know how to start. i need some hint

show that H is subgroup of finite index in Z^n exactly when H is free comutative group of rank n

proove that < x0 , x1 , . . . | [xi , xj ] = 1, i, j, ∈ N_0 ; x0^p = 1 ; (xi) ^ (p ^ i) = x0 , i ∈ N > is not presentation of Prüfer group

proove that if G and H are divisible groups and there is monomorphisms from G to H and from H to G than G and H are isomorphic

characterize primes p and q for which each group of order pq is nilpotent