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<b>let </b><font color="Maroon">G</font> be  <a href="group_1.html#V1">strict</a> <a href="group_1.html#NM1">Group</a>, <font color="Maroon">F</font> be  <a href="group_1.html#V1">strict</a> <a href="group_1.html#NM1">Group</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:26"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">G</font>,<font color="Maroon">F</font> <a href="group_6.html#R2">are_isomorphic</a> 
 <b>and </b><br/><a NAME="E2:26"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">G</font> is  <a href="gr_cy_1.html#V1">cyclic</a> <a href="group_1.html#NM1">Group</a> or <font color="Maroon">F</font> is  <a href="gr_cy_1.html#V1">cyclic</a> <a href="group_1.html#NM1">Group</a> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:26_1_1"/>
( <font color="Maroon">G</font> is  <a href="gr_cy_1.html#V1">cyclic</a> <a href="group_1.html#NM1">Group</a> or <font color="Maroon">F</font> is  <a href="gr_cy_1.html#V1">cyclic</a> <a href="group_1.html#NM1">Group</a> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1"><b>suppose </b></a><a NAME="E1:26_1_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">G</font> is  <a href="gr_cy_1.html#V1">cyclic</a> <a href="group_1.html#NM1">Group</a>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">h</font> being   <a href="group_6.html#NM2">Homomorphism</a> of <font color="Maroon">G</font>,<font color="Maroon">F</font><b> such that </b><br/><a NAME="E3:26_1_1_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">h</font> <a href="group_6.html#NR3" target="_self">is_isomorphism</a> 
 <b>by </b><i>, <a class="ref" href="group_6.html#D15">GROUP_6:def 15</a></i>;<br/><a NAME="E4:26_1_1_1"/>
( <font color="Maroon">h</font> <a href="group_6.html#NR1" target="_self">is_monomorphism</a>  &amp; <font color="Maroon">h</font> <a href="group_6.html#NR2" target="_self">is_epimorphism</a>  )
 <b>by </b><i>, <a class="ref" href="group_6.html#D14">GROUP_6:def 14</a></i>;<br/><a NAME="E5:26_1_1_1"/><b>then </b>
 <a href="group_6.html#K12">Image</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="group_6.html#T67">GROUP_6:67</a></i>;<br/><b>hence </b><a NAME="E6:26_1_1_1"/>
( <font color="Maroon">G</font> is  <a href="gr_cy_1.html#V1">cyclic</a> <a href="group_1.html#NM1">Group</a> &amp; <font color="Maroon">F</font> is  <a href="gr_cy_1.html#V1">cyclic</a> <a href="group_1.html#NM1">Group</a> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_2"><b>suppose </b></a><a NAME="E1:26_1_1_2"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">F</font> is  <a href="gr_cy_1.html#V1">cyclic</a> <a href="group_1.html#NM1">Group</a>
 ;<br/><div class="add"><a NAME="E2:26_1_1_2"/>
<font color="Maroon">F</font>,<font color="Maroon">G</font> <a href="group_6.html#R2">are_isomorphic</a> 
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">h</font> being   <a href="group_6.html#NM2">Homomorphism</a> of <font color="Maroon">F</font>,<font color="Maroon">G</font><b> such that </b><br/><a NAME="E4:26_1_1_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">h</font> <a href="group_6.html#NR3" target="_self">is_isomorphism</a> 
 <b>by </b><i><a class="ref" href="group_6.html#D15">GROUP_6:def 15</a></i>;<br/><a NAME="E5:26_1_1_2"/>
( <font color="Maroon">h</font> <a href="group_6.html#NR1" target="_self">is_monomorphism</a>  &amp; <font color="Maroon">h</font> <a href="group_6.html#NR2" target="_self">is_epimorphism</a>  )
 <b>by </b><i>, <a class="ref" href="group_6.html#D14">GROUP_6:def 14</a></i>;<br/><a NAME="E6:26_1_1_2"/><b>then </b>
 <a href="group_6.html#K12">Image</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="group_6.html#T67">GROUP_6:67</a></i>;<br/><b>hence </b><a NAME="E7:26_1_1_2"/>
( <font color="Maroon">G</font> is  <a href="gr_cy_1.html#V1">cyclic</a> <a href="group_1.html#NM1">Group</a> &amp; <font color="Maroon">F</font> is  <a href="gr_cy_1.html#V1">cyclic</a> <a href="group_1.html#NM1">Group</a> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:26"/>
( <font color="Maroon">G</font> is  <a href="gr_cy_1.html#V1">cyclic</a> <a href="group_1.html#NM1">Group</a> &amp; <font color="Maroon">F</font> is  <a href="gr_cy_1.html#V1">cyclic</a> <a href="group_1.html#NM1">Group</a> )
 ;<br/>


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