Articles of group theory

Isomorphic quotient groups $\frac{G}{H} \cong \frac{G}{K}$ imply $H \cong K$?

I know that given a group $G$ and two normal subgroups $H,K \subset G$ then it is not true that: “if $H \cong K$ then $ \frac{G}{H} \cong \frac{G}{K} $ (the counterexample is quite easy with products of cyclic groups) “ My question is: Is the converse true? i.e. Given that $\frac{G}{H} \cong \frac{G}{K}$ then […]

Nonisomorphic groups of order 12.

I’m trying to find 4 groups of order 12, none of which are isomorphic to each other. Should I be trying external direct products? So far I have $A_4, \mathbb Z_{12},\,$ and $\,\mathbb Z_6\times \mathbb Z_2.\,$ How do I show all of these are non-isomorphic to each other? And how do I find a fourth? […]

Isomorphisms: preserve structure, operation, or order?

Everyone always says that isomorphisms preserve structure… but given the (multiple) definitions of isomorphism, I fail to see how the definitions equate with the intuitive meaning, which is that two sets are “basically the same if you ignore naming and notation”. Here are the different definitions I’ve come across: Order Isomorphism Let $A$ be a […]

$|G|>2$ implies $G$ has non trivial automorphism

Well, this is an exercise problem from Herstein which sounds difficult: How does one prove that if $|G|>2$, then $G$ has non-trivial automorphism? The only thing I know which connects a group with its automorphism is the theorem, $$G/Z(G) \cong \mathcal{I}(G)$$ where $\mathcal{I}(G)$ denotes the Inner- Automorphism of $G$. So for a group with $Z(G)=(e)$, […]

What is the number of invertible $n\times n$ matrices in $\operatorname{GL}_n(F)$?

$F$ is a finite field of order $q$. What is the size of $\operatorname{GL}_n(F)$ ? I am reading Dummit and Foote “Abstract Algebra”. The following formula is given: $(q^n – 1)(q^n – q)\cdots(q^n – q^{n-1})$. The case for $n = 1$ is trivial. I understand that for $n = 2$ the first row of the […]

If $$ and $$ are relatively prime, then $G=HK$

I’m struggling to proof that if $H$ and $K$ are subgroups of finite index of a group $G$ such that $[G:H]$ and $[G:K]$ are relatively prime, then $G=HK$. I don’t know why I can’t answer it, because this question seems easy. I’m stuck maybe because I’ve studied so far just Lagrange’s theorem and some of […]

how to show $SU(2)/\mathbb{Z}_2\cong SO(3)$

Could any one give me hint for this one? how to show $SU(2)/\mathbb{Z}_2\cong SO(3)$, well, Is it the same: there is a 2-fold covering map from $SU(2)$ to $SO(3)$? what is that map will be?

Coloring dodecahedron

I found some months ago that there are the Polya’s enumeration theorem to compute number of colorings of dodecahedron. I got interested to find how to show by using only Burnside’s lemma that there are 9099 ways to color dodecahedrom by three colors. How can I do the computation?

Algebra: Best mental images

I’m curious how people think of Algebras (in the universal sense, i.e., monoids, groups, rings, etc.). Cayley diagrams of groups with few generators are useful for thinking about group actions on itself. I know that a categorical approach is becoming more mainstream. For me, lattice theory is my fallback. Lattice theory is useful to remember […]

Isomorphism between $I_G/I_G^2$ and $G/G'$

Ok, this has been bugging me for a while, and I’m sure there’s something obvious I’m missing. The references I’ve looked at for this result in an effort to resolve the issue didn’t address it. $G$ is a group, $\mathbb{Z}[G]$ its integral group ring, $I_G$ the augmentation ideal (i.e. the kernel of the map $\mathbb{Z}[G]\rightarrow\mathbb{Z}$ […]