Give an example of a non-abelian group $G$ such that $H = \{a \in G \mid\operatorname{ord}(a) < \infty\}$ is not a subgroup of $G$.
8742
Examples and further results about the order of the product of two elements in a group
98542
Prove that $H$ is a subgroup of an abelian group $G$
92493
If $G$ is non-abelian, then $Inn(G)$ is not a normal subgroup of the group of all bijective mappings $G \to G$
109060
Group containing no subgroup of index 2
151047
a non trivial example: $G/H$ is isomorphic to $G$
35401
Examples of finite groups that are not a semidirect product
8066
Sort-of-simple non-Hopfian groups
19996
Example where union of increasing sigma algebras is not a sigma algebra
61084
Examples of non-isomorphic fields with isomorphic group of units and additive group structure
97565
Quotient group $G/H$ is abelian iff $[G,G] \subseteq H$
141043
$H$ normal in $G$. Need $G$ contain a subgroup isomorphic to $G/H$
154398
Show that H is a subgroup of G
107935
If $G$ is a group, $H$ is a subgroup of $G$ and $g\in G$, is it possible that $gHg^{-1} \subset H$?
59902
Example of $2$ non-isomorphic groups that have the same quotients
132704
If G is not commutative, then is there always a subgroup that is not a normal subgroup?
47159
Wanted: example of an increasing sequence of $\sigma$-fields whose union is not a $\sigma$-field
107511
If $G$ and $H$ are nonisomorphic group with same order then can we say that $Aut(G)$ is not isomorphic to $Aut(H)$?