In the paper ‘Nilpotent Numbers’ by Pakianathan and Shankar (http://www2.math.ou.edu/~shankar/papers/nil2.pdf), it was proven that every group of order $n$ is nilpotent if and only if $p^k\not\equiv 1\mod q$ whenever $p,q$ are distinct primes with $p^k$ and $q$ dividing $n$. I am interested in finding conditions on $n$ such that every group of order $n$ is […]

The set $G=\left\{\begin{pmatrix}1 & n \\ 0 & 1 \\ \end{pmatrix}\mid n\in \Bbb Z\right\}$ with the operation of matrix multiplication is a group. Show that $$\phi:\Bbb Z \to G,$$ $$\phi(n)=\begin{pmatrix}1 & n \\ 0 & 1 \\ \end{pmatrix}$$ is a group isomorphism (where the operation on $\Bbb Z$ is ordinary addtion). TO show it’s isomorphism: […]

How can I prove that $$ \operatorname{Aut}(\mathbb {D}_n) \cong \mathbb {Z}_n \rtimes \operatorname{Aut}(\mathbb {Z}_n), $$ where $\mathbb {D}_n$ is the dihedral group. Can someone help me please? Thank you.

Let $G\cong \Bbb{Z}_{p^{r_1}}\oplus\Bbb{Z}_{p^{r_2}}\oplus\cdots \oplus \Bbb{Z}_{p^{r_s}}$ be a finite abelian $p$-group, where $r_1\geq r_2\geq \cdots \geq r_s\geq 1$. Let $H\cong \Bbb{Z}_{p^{t_1}}\oplus\Bbb{Z}_{p^{t_2}}\oplus\cdots \oplus \Bbb{Z}_{p^{t_u}}$ be a subgroup of $G$, where $t_1\geq t_2\geq \cdots \geq t_u\geq 1$. Prove that $s\geq u$ and $r_i\geq t_i$ for each $i=1, 2, …, u$. It seems obviously. But I just can’t prove […]

Let $G$ be Schmidt group. If $Q=\langle a\rangle$ is $q$-subgroup of $G$, then $a^q \in Z(G)$

Prove that $Q_{8} \cong H \rtimes G \Rightarrow H=\{e\}$ or $G=\{e\}$. My proof (is it correct?): We know that (it’s a fact from the lecture): If $M \cong H\rtimes G$, then $H \cong K \unlhd M$ and $G\cong L \leq M$. $K \cap L =\{e\}$. For finite groups: $|K||L|=|M|$. As every subgroup of $Q_{8}$ is […]

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 […]

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