document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=847208>Question 847208</A>: <I><SPAN STYLE=\"word-break: break-all;\"> hello,<BR>\n" );
document.write( "if I have (z,+) as a group , <BR>\n" );
document.write( "h={2x,x belongs z)<BR>\n" );
document.write( "h={-8,-4,-2,0,2,4,8}=2z<BR>\n" );
document.write( "2z C z<BR>\n" );
document.write( "0 belongs to 2z(0=2*0) and h!= empty set<BR>\n" );
document.write( "how can i show that 2z is a subgroup of z ? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #510267 by </B><A HREF=/tutors/aboutme.mpl?userid=swincher4391><B>swincher4391(1107)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=swincher4391><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=swincher4391><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=510267\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Ah, abstract algebra. Good times.\r<BR>\n" );
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document.write( "First off, let's think about how we can show how one set is a subgroup of another.\r<BR>\n" );
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document.write( "We must show closure, identity, and an inverse element. \r<BR>\n" );
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document.write( "Let's recap. We have h = {2x | where x is an integer}\r<BR>\n" );
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document.write( "We want to show that 2z is a subgroup of z under the additive operator.\r<BR>\n" );
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document.write( "Closure under addition:\r<BR>\n" );
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document.write( "Let 2x and 2y be integers.\r<BR>\n" );
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document.write( "Then  2x + 2y = 2(x+y) which exists in 2z.\r<BR>\n" );
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document.write( "Identity:\r<BR>\n" );
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document.write( "We know that in Z the identity element is 0. Since we have that 0 = 2*0,  0 exists in 2z.\r<BR>\n" );
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document.write( "Inverse: \r<BR>\n" );
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document.write( "Suppose we have some 2x in Z. The additive inverse for 2x is -2x and  -2x = 2*(-x) which means -2x exists in 2z\r<BR>\n" );
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document.write( "Hence 2z is a subgroup of z under addition.\r<BR>\n" );
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document.write( "Note the upshot is we try to find elements that have the form  2 * (something).\r<BR>\n" );
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document.write( " </SPAN>\n" );
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