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Prove that the following are equivalent ~ is an equivalence relation of a group G
If H and K are normal subgroups of a group G with HK = G. Prove that G/(H n K) = (G/H) x (G/K).
Let G be a group, let a, b be elements in G and let m and n be (not necessarily positive) integers. (a^n)^m= (a^mn)
Give an example of groups Hi, Kj such that H1xH2 is isomorphic to K1xK2 and no Hi is isomorphic to any Kj.
Prove that if G is a finite group, H subset of G that is closed with respect to the operation of G, Then every element of H has its inverse in H.
Prove that there cannot be a nilpotent group N generated by two elements with the property that every nilpotent group generated by two elements.
Let G be a nonempty finite set equipped with an associative operation such that for all a,b,c,d in G. Prove that G is a group.
Prove that the cyclic subgroup generated by a^m is the same as the cyclic subgroup generated by a^d, where d = (m,n).
Show that G is a group of order 8 and that G is isomorphic to the quaterunion group Q = {1, i, -1, -i, j, k, -j, -k }.
Prove that the subgroup of A4 generated by any element of order 2 and and any element of order 3 is all of A4.
If G is nonabelian and the order of G is 24 and G is isomorphic to H x Z_3, what are the possibilities for H up to isomorphism.
Let P be a normal Sylow p-subgroup of G and let H be any subgroup of G. Prove that P intersect H is the unique Sylow p-subgroup of H.
Show that a group G of order 2p^n has proper normal subgroup, where p is odd prime number and n > 0.
Using examples and/or theorems, argue that G has at least one subgroup of every order dividing |G|.
Let G be a group with A and B subgroups of G. Prove that the set AB = {ab | a is in A and b is in B} is a subgroup of G if and only if AB = BA
Let G be a group and S any subset of G. Prove that C_G (S) = {g in G such that gs = sg for all s in S} is a subgroup of G.
G1=G2=Z5 (the integers modulo 5 with the operation +), f:n?an(mod5) where a?Z_5\{0}={1,2,3,4} is fixed.
Give two reasons why the set of odd integers would not form a group under ordinary addition.
When an element and its inverse are combined under and operation the result is the identity element. The identity element for multiplication is 1.
Let i be an integer with 1 <= i <= n. Let Gi* be the subset of G1 X ... X Gn consisting of those elements whose ith coorinate is any element of Gi.
Each one need only be 1 to just a few sentences - please list separately and under each fact provide any links to websites or pictures etc.
Find all the integers such that when the final digit is deleted the new integer divides the original one.
A = U X U, where U = {1, 2, 3, 4, 6, 12}, and %:A -> Q is defined by %(n, m) = n/m, where Q is the set of all rational numbers.
Let G = GL(2,R) be the general linear group. Let H=GL(2,Q) and K= SL(2,R) = {A is an element of G: det (A) =1} Show that H,K are subgroups of G.