As shown in this note, the symmetry group $S_4$ for a cube has $3$ subgroups that are isomorphic to $D_4$, the dihedral group of order $2 \times 4 = 8$. How to geometrically illustrate this fact? Specifically, where are the squares embedded in the cube? Related post: How to geometrically show that there are $4$ […]

I’m interested in the following exercise from Dummut & Foote’s Abstract algebra text (p. 151) Let $D$ be the subgroup of $S_\Omega$ consisting of permutations which move only a finite number of elements of $\Omega$ (described in Exercise 17 in Section 3) and let $A$ be the set of all elements $\sigma \in D$ such […]

Show that the symmetric group $S_4$ has a subgroup of order $d$ for each $d|24$. From Lagrange’s theorem I know that if $G \le S_4$, then the order of $G$ necessarily divides $|S_4|=24$. However the question actually asks the converse of the Lagrange’s theorem, so I cannot apply the theorem directly. (And I don’t think […]

I am going through Quantum factoring, discrete logarithms and the hidden subgroup problem by Richard Jozsa. On page 13, the author discussed the hidden subgroup problem (HSP) formulation of the graph isomorphism (GI) problem. I would like to make it sure that I get the development of the concept right. Here both $A$ and $B$ […]

I know that the symmetric group $S_n$ is generated by $(12)$ and $(2345\dots n)$. Let $G$ be a transitive subgroup of $S_n$ (transitive with respect to the natural action of $S_n$ on $\{1,2,\dots,n\}$) that contains a transposition and an $(n-1)$-cycle. Prove that $G=S_n$.

During reading a book, I have faced to this problem telling: $G$ is a group of order $12$ such that $Z(G)$ has no element of order $2$ . Then $G≅A_4$. Obviously, this group is not abelian and I think some information about $S_4$ is involved here because of the desired deduction. Can we say $|\frac{G}{Z(G)}|≠3$? […]

I use Abstract Algebra by Dummit and Foote to study abstract algebra! At page 120, section 2 in chapter 4, there is a great result form my point of view which proves that, for any group $G$ of order $n$, $G$ is isomorphic to some subgroup of $S_n$. My question: Is there any way to […]

The subgroup in $S_4$ that I know has order 12 is the subgroup of all even permutations, otherwise known as the alternating group $A_4$. However, I know this from a fact and not because I am able to show a subgroup of order 12 exists in $S_4$ in the first place. If I had not […]

What is the “Path” for understanding what Frieze Groups really are? Generally in mathematics, there is a is a path or “building blocks” approach to learning something. For example if I know how to count and I want to learn how to multiply you say the path is “Counting -> Adding (adding is fast counting) […]

I’m working on the following question, and honestly have no idea how to begin. Any hints would be greatly appreciated! Let $H$ be a subgroup of $S_n$, the symmetry group of the set $\{1,2,\dots, n\}$. Show that if $H$ is transitive and if $H$ is generated by some set of transpositions, then $H=S_n$.

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