Questions tagged [group-theory]

Question 1

We have $F$ a free group. If we have a chain $F = C_0 \gt C_1 \gt \ldots$ such that each $C_{i+1}$ is a proper characteristic subgroup of $C_i$. We want to prove $\cap_i C_i = 1$. Now, I have a ...

Question 2

Let $G$ be a non-abelian simple group of order $60$. Then $G \cong A_5$. Prove the uniqueness. The third Sylow theorem should be used. $|G|=60=2^2⋅3⋅5$ $2k+1|60$ I am analyzing all possibilities.

Question 3

Let $A = \left\{ 1, 2, 3, \ldots, n \right\}$. Let $B$ be the set of all bijections from $A$ to $A$ (i.e., the symmetric group $S_n$). Let $C$ be a non-empty subset of $B$ such that for every ...

Question 4

In "A Course in the Theory of Groups" written by Derek J.S. Robinson it is done as follows. Let be $X$ a set. Choose a set disjoint from $X$ with the same cardinality (see here for details):...

Question 5

In the lecture of group theory and Lie algebra we were told: For every $A\in \cal{A}$ we define the linear map $ad_A$: $ad_A: X\in \cal{A}$ $\rightarrow$ $ad_A X=[X,A]\in \cal{A}$. The adjoin ...

Question 6

Let $n\in \mathbb{N}$ with $n\geq 2$ and let $S_n$ be the symmetric group of degree $n$. Let $I:= \{1,\cdots,n\}$ and let $\alpha$ and $\beta$ be two transpositions of $I$ defined by $\alpha(a)=b$, $\...

Question 7

I understand that the only difference between left and right group actions is the order in which gh acts on x. Since cosets Hg and gH involve the action of one element of the subgroup H on one element ...

Question 8

Thus, for establishing general properties of group actions, it suffices to consider only left actions. However, there are cases where this is not possible. For example, the multiplication of a group ...

Question 9

Conjecture. Let $G$ be a group and $B$ any set of generators for $G$. That is to say $G = (G, \cdot) = \langle B \rangle$. Then for any equation $E=F$ in $G$ that is constant-free, we have that $E=...

Question 10

I just constructed a group of order 5 that is not abelian. Clearly, something is theoretically wrong, but I can't figure out what.

Question 11

If $H$ is a subgroup of a group $(G,\ast,e)$ then it is said pronormal iff for any $g$ in $G$ there exists $x$ in $\left\langle H\cup(g\ast H\ast g^{-1})\right\rangle$ such that the equality $$ g\ast ...

Question 12

I'm currently studying different aspects of the braid group, and I've come across various different definitions. In particular, I don't understand what is the connection between the pure braid group (...

Question 13

If $H$ and $K$ are subgroup of a group $(G,\ast,e)$ then I know that $H\ast K$ is a subgroup of when $H$ is commutable with $K$: so I am searching a counterexample showing that if a subgroup $X$ ...

Question 14

As the title indicates, I'm trying to prove that, when $o(G) = p^3$ for $p$ an odd prime, $G$ must be regular, i.e. for any $a, b \in G$, $(ab)^p = a^pb^pc^p$ for some $c \in \langle a, b \rangle^1$. ...

Question 15

Let us consider the algebraic group $G=\mathrm{GL}_2(\mathbb{C})$ and consider the $S_2$-action given by conjugation with $P_0=\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$, that is, the $S_2$-...

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Questions tagged [group-theory]

If we have a Free Group with a chain of proper characteristic subgroups, then their intersection is trivial

Let $G$ be a non-abelian simple group of order $60$. Then $G \cong A_5$. Prove the uniqueness. [duplicate]

A problem about permutation group to find the minimum cardinality of a set

Robinson's word vs Rotman's word: can the second be derived from the first?

Why elements of a (Lie) algebra act on the algebra, in the adjoint representation? [closed]

Transpositions in the symmetric group are conjugated elements of the group [duplicate]

Why are left and right cosets different in the non-normal case?

How are left/right multiplication different from other group actions?

Conjecture: a constant-free equation is solvable in a group $G$ if and only if it is solvable in its generating set $B$ for any $G=\langle B\rangle$

Non-abelian group of order five? [closed]

If $H$ and $N$ are respectively a pronormal and normal subgroup then is $H\cap N$ pronormal in $N$?

Infinitesimal vs pure braid relations

If $H$, $K$ and $L$ are subgroup such that $H$ commutes with $K$ and $L$ then is $H\ast K\ast L$ a subgroup?

Showing order $p^3$ groups are regular for odd $p$

Quotient of $\mathrm{GL}_2(\mathbb{C})$ by a finite group

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