Some Examples and Counterexamples in Group Theory

This is a collection of groups that have specific interesting properties, such as being a counterexample to some non-obvious statements about groups. Most of these are well known.

• A group isomorphic to every non-trivial subgroup. $\mathbb{Z}$.
• A group of order $n$ which has no subgroup of $k$, where $k \mid n$. $A_5$, since it is simple and thus has no subgroup of index 2. Another example is $A_4$ (which is the smallest such group), but it’s a little harder to show that it works.
• Two non-isomorphic groups with the same order type. $C_4 \times C_4$ and $C_2 \times Q_8$. Both have 1 element of order 1, 3 elements of order 2, and 12 elements of order 4. If this property holds for two abelian groups, then they are isomorphic.
• A non-abelian group with all non-identity elements of order $p$. The group of upper triangular $3 \times 3$ matrices over $\mathbb{Z}/p\mathbb{Z}$ with 1s on the diagonal.
• A non-abelian group of order $p^3$. Same as above.
• An infinite group whose proper subgroups are all finite. The group $\left\{k/2^{n} : k, n \in \mathbb{N}, k<2^{n}\right\}$ with addition modulo 1. This is the Prüfer $2$-group.
• A group $G$ with $N \trianglelefteq G$ and $H \trianglelefteq N$ such that $H \not \trianglelefteq G$. In $D_8$, we have $\langle s \rangle \trianglelefteq \langle r^2, s \rangle \trianglelefteq \langle r, s \rangle = D_4$, but $\langle s \rangle \not \trianglelefteq D_4$.
• A group in which every group generated by $n$ elements has a surjective homomorphism to it. The free group with a basis of $n$ elements.
• An infinite group with every non-identity element of order 2. $C_2 \times C_2 \times C_2 \times \cdots$.
• An infinite non-abelian group with every element of finite order. $S_3 \times C_2 \times C_2 \times \cdots$.
• A group $G$ with $G \cong G \times G$. $G = C_2 \times C_2 \times C_2 \times \cdots$. We can also get $G \cong G \times G \times G \times \cdots$ in the natural way.
• Two non-zero elements of $\mathbb{R}$ that generate a subgroup not isomorphic to $\mathbb{Z}$. $1$ and $\sqrt{2}$.
• A group with two subgroups whose product is not a subgroup. Consider $D_3$, with the distinct reflections $s, s’ \in D_3$. Then ${e, s} \times {e, s’} \not \leq D_3$.
• A quotient group of a finite group that is not isomorphic to a subgroup. $Q_8 / {-1, 1}$.
• A group isomorphic to its automorphism group. $S_3 \cong \operatorname{Aut}(S_3)$.
• A non-abelian infinite group where the set of elements of finite order is a subgroup. $\operatorname{GL}_2(\mathbb{Q})$. We note that this also holds for all abelian groups, and indeed any group where the elements of finite order commute with each other.
• A group that is not a semi-direct product. $Q_8$.
• A group $G$ with two isomorphic subgroups $H, K$ where $G/H$ and $G/K$ are not isomorphic. Take $G = C_4 \times C_2$, with $H = \langle (2, 0) \rangle$ and $K = \langle (0, 1)\rangle$. Then $G/H \cong C_2 \times C_2$, and $G/K \cong C_4$.