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 […]

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 […]

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.

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 […]

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 […]

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 […]

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 […]

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 […]

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 […]

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.

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