document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=14132>Question 14132</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 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. <BR>\n" );
document.write( "I have one example but I need more\r<BR>\n" );
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document.write( "G= group of all 2x2 matrices under addition. <BR>\n" );
document.write( "A= group of all 2x2 matrices under addition having first element of first row as non-zero while all the other three are zero.<BR>\n" );
document.write( "B= group of all 2x2 matrices under addition having second element of first row as non-zero while all the other three are zero.<BR>\n" );
document.write( "c= group of all 2x2 matrices under addition having first element of second row as non-zero while all the other three are zero.<BR>\n" );
document.write( "D= group of all 2x2 matrices under addition having second element of second row as non-zero while all the other three are zero.\r<BR>\n" );
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document.write( "then G = AxBxCxD  that is G is direct product of A,B,C,D. Please give me such more examples.Thankyou.<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #7090 by </B><A HREF=/tutors/aboutme.mpl?userid=khwang><B>khwang(438)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=khwang><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=7090\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">  The decomposition of the given group G is called direct product if<BR>\n" );
document.write( " G = H x K for subgroups H and K of G.\r<BR>\n" );
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document.write( " In case of abelian, G is called the direct sum of H & K and denoted<BR>\n" );
document.write( " by H + K.\r<BR>\n" );
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document.write( " The example that you gave is merely OK, since the form you wrote was<BR>\n" );
document.write( " not clear and not in good shapes.<BR>\n" );
document.write( "  <BR>\n" );
document.write( " Just close your eyes, there are thousands of such examples as<BR>\n" );
document.write( " R^2 = R + R, R^3 = R^2 + R. \r<BR>\n" );
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document.write( " Your example should express what your addition group G of  2x2 matrices over Z,<BR>\n" );
document.write( " over Q or R or C (whatever)<BR>\n" );
document.write( "  If G is the addition group of 2x2 matrices over Z.<BR>\n" );
document.write( " Since dim G = 4, let H = {[a b]<BR>\n" );
document.write( "                            [0 0] | a,b in Z}<BR>\n" );
document.write( " and K = {[0 0]<BR>\n" );
document.write( "          [c d] | c,d in Z}\r<BR>\n" );
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document.write( " then G = H x K [Note dim H = dim K= 2]\r<BR>\n" );
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document.write( " For finite group , let G = Z6 (i.e Z6 = Z/6Z,mod group of Z )<BR>\n" );
document.write( " H = {[0],[2],[4]} , K = {[0],[3]}<BR>\n" );
document.write( " then G = H x K. (why ?)\r<BR>\n" );
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document.write( " In general, Zmn is isomorphic to Zm x Zn if m & n are relative prime.\r<BR>\n" );
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document.write( " Try to look for more examples in the web or books.\r<BR>\n" );
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document.write( " Of course, you have to work hard.\r<BR>\n" );
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document.write( " Kenny\r<BR>\n" );
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document.write( "  <BR>\n" );
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