Residually finite group

In the <a href="/facts/Mathematics/pxTouaz4">mathematical</a> field of <a href="/facts/Group_theory/NfowcI5H">group theory</a>, a <a href="/facts/Group_(mathematics)/IQTYqINt">group</a> G is residually finite or finitely approximable if for every element g that is not the identity in G there is a <a href="/facts/Group_homomorphism/sDguaNYs">homomorphism</a> h from G to a <a href="/facts/Finite_group/vv2tvogB">finite group</a>, such that

h
 (
 g
 )
 ≠
 1.
 
 
 
 {\displaystyle h(g)\neq 1.\,}
 
<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a>
There are a number of equivalent definitions:

<ul><li>A group is residually finite if for each non-identity element in the group, there is a <a href="/facts/Normal_subgroup/aRDaGwLm">normal subgroup</a> of finite <a href="/facts/Index_(group_theory)/ewY2A71Y">index</a> not containing that element.</li>
<li>A group is residually finite if and only if the intersection of all its <a href="/facts/Subgroup/r91mBgpa">subgroups</a> of finite index is <a href="/facts/Trivial_group/XPjQkQFm">trivial</a>.</li>
<li>A group is residually finite if and only if the intersection of all its normal subgroups of finite index is trivial.</li>
<li>A group is residually finite if and only if it can be embedded inside the <a href="/facts/Direct_product_of_groups/4MWCxY0I">direct product</a> of a family of finite groups.</li></ul>

Residually finite group open-in-new

Residually finite group