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<b>let </b><font color="Maroon" title="c1">G</font>, <font color="Maroon" title="c2">F</font> be  <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a> <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:26"/><i><font color="Green" title="E19">A1</font></i>: 
<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">F</font> <a href="group_6.html#R2" title="GROUP_6:pred.2">are_isomorphic</a> 
 <b>and </b><br/><a NAME="E2:26"/><i><font color="Green" title="E20">A2</font></i>: 
( <font color="Maroon" title="c1">G</font> is  <a href="gr_cy_1.html#V1" title="GR_CY_1:attr.1">cyclic</a> <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a> or <font color="Maroon" title="c2">F</font> is  <a href="gr_cy_1.html#V1" title="GR_CY_1:attr.1">cyclic</a> <a href="group_1.html#NM1" title="GROUP_1:NM.1">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" title="c1">G</font> is  <a href="gr_cy_1.html#V1" title="GR_CY_1:attr.1">cyclic</a> <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a> or <font color="Maroon" title="c2">F</font> is  <a href="gr_cy_1.html#V1" title="GR_CY_1:attr.1">cyclic</a> <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a> )
 </span><b>by </b><i><a class="txt" href="gr_cy_2.html#E2:26"><i><font color="Green" title="E20">A2</font></i></a></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" title="E21">A3</font></i>: 
<font color="Maroon" title="c1">G</font> is  <a href="gr_cy_1.html#V1" title="GR_CY_1:attr.1">cyclic</a> <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c3">h</font> being   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">F</font><b> such that </b><br/><a NAME="E3:26_1_1_1"/><i><font color="Green" title="E22">A4</font></i>: 
<font color="Maroon" title="c3">h</font> <a href="group_6.html#NR3" title="GROUP_6:NR.3">is_isomorphism</a> 
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E1:26"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="group_6.html#D15" title="GROUP_6:def.15">GROUP_6:def 15</a></i>;<br/><a NAME="E4:26_1_1_1"/>
( <font color="Maroon" title="c3">h</font> <a href="group_6.html#NR1" title="GROUP_6:NR.1">is_monomorphism</a>  &amp; <font color="Maroon" title="c3">h</font> <a href="group_6.html#NR2" title="GROUP_6:NR.2">is_epimorphism</a>  )
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E3:26_1_1_1"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="group_6.html#D14" title="GROUP_6:def.14">GROUP_6:def 14</a></i>;<br/><a NAME="E5:26_1_1_1"/><b>then </b>
 <a href="group_6.html#K12" title="GROUP_6:func.12">Image</a> <font color="Maroon" title="c3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="ref" href="group_6.html#T67" title="GROUP_6:th.67">GROUP_6:67</a></i>;<br/><b>hence </b><a NAME="E6:26_1_1_1"/>
( <font color="Maroon" title="c1">G</font> is  <a href="gr_cy_1.html#V1" title="GR_CY_1:attr.1">cyclic</a> <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a> &amp; <font color="Maroon" title="c2">F</font> is  <a href="gr_cy_1.html#V1" title="GR_CY_1:attr.1">cyclic</a> <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a> )
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E1:26_1_1_1"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="gr_cy_2.html#T32" target="_self" title="GR_CY_2:th.32">Th32</a></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" title="E21">A5</font></i>: 
<font color="Maroon" title="c2">F</font> is  <a href="gr_cy_1.html#V1" title="GR_CY_1:attr.1">cyclic</a> <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>
 ;<br/><div class="add"><a NAME="E2:26_1_1_2"/>
<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c1">G</font> <a href="group_6.html#R2" title="GROUP_6:pred.2">are_isomorphic</a> 
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E1:26"><i><font color="Green" title="E19">A1</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c3">h</font> being   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E4:26_1_1_2"/><i><font color="Green" title="E22">A6</font></i>: 
<font color="Maroon" title="c3">h</font> <a href="group_6.html#NR3" title="GROUP_6:NR.3">is_isomorphism</a> 
 <b>by </b><i><a class="ref" href="group_6.html#D15" title="GROUP_6:def.15">GROUP_6:def 15</a></i>;<br/><a NAME="E5:26_1_1_2"/>
( <font color="Maroon" title="c3">h</font> <a href="group_6.html#NR1" title="GROUP_6:NR.1">is_monomorphism</a>  &amp; <font color="Maroon" title="c3">h</font> <a href="group_6.html#NR2" title="GROUP_6:NR.2">is_epimorphism</a>  )
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E4:26_1_1_2"><i><font color="Green" title="E22">A6</font></i></a>, <a class="ref" href="group_6.html#D14" title="GROUP_6:def.14">GROUP_6:def 14</a></i>;<br/><a NAME="E6:26_1_1_2"/><b>then </b>
 <a href="group_6.html#K12" title="GROUP_6:func.12">Image</a> <font color="Maroon" title="c3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="ref" href="group_6.html#T67" title="GROUP_6:th.67">GROUP_6:67</a></i>;<br/><b>hence </b><a NAME="E7:26_1_1_2"/>
( <font color="Maroon" title="c1">G</font> is  <a href="gr_cy_1.html#V1" title="GR_CY_1:attr.1">cyclic</a> <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a> &amp; <font color="Maroon" title="c2">F</font> is  <a href="gr_cy_1.html#V1" title="GR_CY_1:attr.1">cyclic</a> <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a> )
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E1:26_1_1_2"><i><font color="Green" title="E21">A5</font></i></a>, <a class="ref" href="gr_cy_2.html#T32" target="_self" title="GR_CY_2:th.32">Th32</a></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" title="c1">G</font> is  <a href="gr_cy_1.html#V1" title="GR_CY_1:attr.1">cyclic</a> <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a> &amp; <font color="Maroon" title="c2">F</font> is  <a href="gr_cy_1.html#V1" title="GR_CY_1:attr.1">cyclic</a> <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a> )
 ;<br/>


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