Articles of group theory

Confused by Example in Herstein's “Topics in Algebra”

The following comes from I.N. Herstein’s “Topics in Algebra”, just after defining subgroups. He gives the following example Let $S$ be any set and $A(S)$ be the set of one-to-one mappings of $S$ onto itself, made into a group under the composition of mappings. For any $x_0 \in S$ define $H(x_0) = \{ \phi \in […]

An exact sequence of unit groups

In the answer of K. Conrad to this question, he mentions a “nice 4-term short exact sequence of abelian groups (involving units groups mod a, mod b, and mod ab)” proving the product formula for $\phi$. How does this sequence look like? (I couldn’t figure it out myself, the term $(a,b)$ in the formula puzzles […]

Order of a center of a group is prime order

Question : Suppose that $G$ is a non-abelian group of order $p^{3}$ where $p$ is prime and $Z (G) \neq \{e\}$. Prove that $|Z (G)| =p$. Any useful hint to this question is appreciated. Thanks in advance.

An epimorphism in $\text{Grp}$ without right inverse?

Exercise 8.24 in Aluffi’s Algebra: Chapter 0 asks us to find an epimorphism in $\text{Grp}$ without right inverses. I happen to know that epimorphisms in $\text{Grp}$ are surjective, so we need a surjective $G\xrightarrow{\phi}G’$ without right inverses. But $G’\cong G/\operatorname{ker}(\phi)$, so in a sense we need a $G/\operatorname{ker}(\phi)$ that cannot be realised as a subgroup […]

A problem from Dummit Foote on Representation of finite groups over algebraic closure of $\mathbb{Q}$

Let $\bar{\mathbb{Q}}$ be the algebraic closure of $\mathbb{Q}$ in $\mathbb{C}$.Let $\phi:G \rightarrow Gl_{m}(\bar{\mathbb{Q}})$ be an irreducible representation. Then $\phi$ can be viewed as a representation from G to $Gl_{m}(\mathbb{C})$. Prove the following: 1) $\phi$ viewed as a representation of G to $Gl_{m}(\mathbb{C})$ is irreducible. 2)Show that the set of irreducible characters over $\bar{\mathbb{Q}}$ is same […]

Are there automorphisms of $H$ which are not restriction of an automorphism of $G$?

Let $G$ be a group and $H$ a characteristic subgroup of $G$ (that is, invariant under all automorphisms of $G$). Let $\phi \in \mathcal{Aut}(H)$, we’ll call $\widetilde{\phi}$ an automorphism of $G$ such that $$\widetilde{\phi}(h) = \phi(h) \quad \forall h \in H$$ Or in other words, such that $\widetilde{\phi}_H = \phi$, where $\widetilde{\phi}_H$ denotes its restriction […]

Quotient Objects in $\mathsf{Grp}$

I don’t know how to precisely formulate my question, but here goes: Subobjects and quotient objects are duals, so a quotient object in $\mathsf{Grp}^{\text{op}}$ is a subobject in $\mathsf{Grp}$. The arrows of these categories are in bijection, and the subobjects of a group $G$ in $\mathsf{Grp}$ are in bijection with the subgroups of $G$. I’m […]

How to prove $|S||T|\leq |S \cap T ||\langle S, T\rangle|$?

If $S$ and $T$ are subgroups of a finite group $G$, prove that $|S||T|\leq |S \cap T||\langle S, T\rangle|$. My approach is: since $\langle S, T\rangle$ is the smallest subgroup containg $S$ and $T$, it should at least contain all the elements of the form $st$ where $s$ belongs to $S$ and $t$ belongs to […]

finite group whose only automorphism is identity map

Question is to prove that : A finite group whose only automorphism is identity map must have order at most $2$. What i have tried is : As any automorphism is trivial, so would be inner automorphism i.e., each map for fixed $g\in G $ with $\eta : G\rightarrow G$ taking $h$ to $ghg^{-1}$ is […]

If $H \leq G$ has finite index, then $G$ has a normal subgroup of finite index

From a practice test for a Masters Qual. Exam: Let $H$ be a subgroup of finite index in a group $G$. Prove that $G$ has a normal subgroup $K$ of finite index with $K \subseteq H$. It makes perfect sense intuitively that $K$ should exist, I just don’t know how to get ahold of it.