document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=403583>Question 403583</A>: <I><SPAN STYLE=\"word-break: break-all;\"> This is my question:<BR>\n" );
document.write( "Suppose that G is a group and g,h are elements of G. There exists a k in G such that kgk=h if and only if gh=m^2 for some m in G.\r<BR>\n" );
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document.write( "I have already have:\r<BR>\n" );
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document.write( "Let k be in G and kgk=h. We can perform the operation on each side of the equation to get gkgk=gh=(gk)^2. m=gk and by the closure component of a group we know that gk is in G so we know that m is in G also. \r<BR>\n" );
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document.write( "I do not get how to prove it the other way. Please can you help.\r<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 #285380 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><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/cgi-bin/show-face.mpl?user=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=285380\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> You have proved the sufficiency part, or the (==>) part.\r<BR>\n" );
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document.write( "To prove the necessity part, or the (<==), we proceed as follows:\r<BR>\n" );
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document.write( "Assume <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=gh++=m%5E2 ALIGN=MIDDLE  ALT=\"gh++=m%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > for some m in G. We have to find to find k such that k*g*k = h.\r<BR>\n" );
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document.write( "Now <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=gh++=m%5E2 ALIGN=MIDDLE  ALT=\"gh++=m%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5E%28-1%29%28gh%29m%5E%28-1%29+=+1%5BG%5D ALIGN=MIDDLE  ALT=\"m%5E%28-1%29%28gh%29m%5E%28-1%29+=+1%5BG%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  (We can do this by the existence of the inverse element in G, as well the existence of identity element <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%5BG%5D ALIGN=MIDDLE  ALT=\"1%5BG%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >).<BR>\n" );
document.write( "Left multiply both sides by h, to get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=hm%5E%28-1%29%28gh%29m%5E%28-1%29+=+h ALIGN=MIDDLE  ALT=\"hm%5E%28-1%29%28gh%29m%5E%28-1%29+=+h\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "By associativity,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28hm%5E%28-1%29%29g%28hm%5E%28-1%29%29+=+h ALIGN=MIDDLE  ALT=\"%28hm%5E%28-1%29%29g%28hm%5E%28-1%29%29+=+h\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "Therefore k exists, and is equal to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=hm%5E%28-1%29 ALIGN=MIDDLE  ALT=\"hm%5E%28-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. </SPAN>\n" );
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