SOLUTION: Let G be a group Under a binary operation "*" Having subgroups H and K such that HxK= G. I need some examples on this type of groups. I have one example but I need more G= gro

Question 14132: Let G be a group Under a binary operation "*" Having subgroups H and K such that HxK= G. I need some examples on this type of groups.
I have one example but I need more
G= group of all 2x2 matrices under addition.
A= group of all 2x2 matrices under addition having first element of first row as non-zero while all the other three are zero.
B= group of all 2x2 matrices under addition having second element of first row as non-zero while all the other three are zero.
c= group of all 2x2 matrices under addition having first element of second row as non-zero while all the other three are zero.
D= group of all 2x2 matrices under addition having second element of second row as non-zero while all the other three are zero.
then G = AxBxCxD that is G is direct product of A,B,C,D. Please give me such more examples.Thankyou.

Answer by khwang(438) (Show Source): You can put this solution on YOUR website!
The decomposition of the given group G is called direct product if
G = H x K for subgroups H and K of G.
In case of abelian, G is called the direct sum of H & K and denoted
by H + K.
The example that you gave is merely OK, since the form you wrote was
not clear and not in good shapes.

Just close your eyes, there are thousands of such examples as
R^2 = R + R, R^3 = R^2 + R.
Your example should express what your addition group G of 2x2 matrices over Z,
over Q or R or C (whatever)
If G is the addition group of 2x2 matrices over Z.
Since dim G = 4, let H = {[a b]
[0 0] | a,b in Z}
and K = {[0 0]
[c d] | c,d in Z}
then G = H x K [Note dim H = dim K= 2]
For finite group , let G = Z6 (i.e Z6 = Z/6Z,mod group of Z )
H = {[0],[2],[4]} , K = {[0],[3]}
then G = H x K. (why ?)
In general, Zmn is isomorphic to Zm x Zn if m & n are relative prime.
Try to look for more examples in the web or books.
Of course, you have to work hard.
Kenny

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