A group is an algebraic structure consisting of a set of elements together with an operation that satisfies four conditions: closure, associativity, identity and invertibility. Group theory studies groups.

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Prove A Group is Not Simple

I ran across this problem. Suppose that $G$ is a group where any two elements that are conjugate commute with each other. Then $G$ is not simple. It goes on to state that, in fact, $G$ must be ...
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7answers
342 views

What's a group whose group of automorphisms is non-abelian?

I recently attended an interview for admission to graduate programs in Mathematics. The interviewing professor asked me a question - Tell me a group whose group of automorphisms is non-abelian. ...
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1answer
14 views

Irreducible Representation second rank tensor [on hold]

The rank-2 tensor xixj, where xi are the Cartesian coordinates of the position vectors in three dimensions, has 6 independent elements. Under rotation, these 6 elements decompose into irreducible sets ...
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33 views

Factorizations of Finite Abelian Groups

Every finite abelian group $G$ can be uniquely written as $$\mathbb{Z}/{d_1\mathbb{Z}} \times \mathbb{Z}/{d_2\mathbb{Z}} \times \cdots \times \mathbb{Z}/{d_r\mathbb{Z}},$$ where $d_i$ divides ...
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0answers
8 views

Convolution of measures on a measurable group is associative

I've come across a statement in Kallenberg's Foundations of Modern Probability which claims this and only tells me to use Fubini's theorem. I am not very familiar with this topic and the text doesn't ...
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1answer
36 views

Example of subgroup of $\mathbb Q$ which is not finitely generated

I was looking for the proper subgroup of $\mathbb Q$ which is not finitely generated under the addition operation. We know every finitely generated subgroups of $\mathbb Q$ is finitely cyclic. For ...
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1answer
45 views

What do these group theory notations mean: $\overline{3}\otimes\overline{2}$, $\overline{2}\oplus\overline{3}$

Can you explain or give a good reference to explain notations like $$\Large\overline{3}\otimes\overline{2}\qquad\qquad \overline{2}\oplus\overline{3}$$ and combinations of these. Thank you.
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1answer
22 views

Explicitly decompose $\mathbb{C}^3$ into irreducible representations of $S_3$.

Consider the permutation representation of $S_3$ acting by permuting the elements of a basis of $\mathbb{C}^3$. Explicitly decompose $\mathbb{C}^3$ into irreducible representations. Can someone ...
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1answer
26 views

Find all the possible pairs of submodules $M_1$ and $M_2$ of the $\mathbb{Z}$-module $\mathbb{Z}_{18}$ so that $\mathbb{Z}_{18} = M_1 \oplus M_2$

I started by considering the possible proper subgroups of $\mathbb{Z}_{18}$, which are $\langle\bar{9}\rangle$, $\langle\bar6\rangle$, $\langle\bar3\rangle$ and $\langle\bar2\rangle$, which are also ...
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3answers
54 views

Prove that $G=\{z \in \mathbb{C}: |z|=1\}$ is an abelian group [on hold]

Prove that $G=\{z \in \mathbb{C}: |z|=1\}$ is an Abelian group with the multiplication operation of complex numbers.
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1answer
39 views

Cyclotomic polynomials, properties.

Let $F$ be a field of characteristic prime to $n$, and let $F^a$ be an algebraic closure of $F$. Let $\zeta$ be a primitive $n$th root of unity in $F^a$. I know that the monic polynomial $\Phi_n(X)$ ...
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2answers
54 views

Let $G,*$ a group and $a,b,c,d \in G$. Prove that …

Let $G,*$ a group and $a,b,c \in G$. Prove that the equation $x*a*x*b=x*c$ it has a unique solution in $G$. Ideas? I do not know where to start. D =
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0answers
47 views

Permutation of cosets

Let $G$ be a finite group and $\gamma \in Sym(G)$, such that $\gamma (1) = 1$ and $\gamma (gH) = \gamma (g)H$ for all $g\in G$, $H\leq G$. This means $\gamma$ induces a permutation of the left cosets ...
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2answers
30 views

Issue with associativity of group

Given $G=(1,2)\subset R$ and the operation $x∗y = \frac{3xy-4x-4y+6}{2xy-3x-3y+5}$ Prove that $(G,∗)$ is an abelian group. So here's my issue with this. For it to be a group I must prove that: ...
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1answer
28 views

Sets of compositions of homomorphisms

I am looking of a relation in the form: $$ Hom(X,Z) = Hom(X,Y)\otimes Hom(Y,Z), $$ or: $$ Hom(X,Z) \subseteq Hom(X,Y)\otimes Hom(Y,Z), $$ or similar (maybe it's not a tensor product? maybe the ...
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1answer
26 views

Relations between $R^fG$ and either $\mathbb{C}^fG$ or $\mathbb{Z}^fG$.

Denote by $RG$ the group ring of the group $G$ over the commutative ring $R$. A result by Passman saying that if $R$ is a commutative ring then $$RG=R\otimes_{\mathbb{Z}}\mathbb{Z}G.$$ As a result, ...
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0answers
33 views

Are there groups of order $p^4q^2$ which are not semi-direct product?

It is easy to show that if $G$ is a group of order $p^2q^2$, where $p,q$ are primes with correspondings Sylow subgroups $P,Q$, that $G$ is a semi-direct product of $P$ and $Q$. Moreover, if $pq\neq ...
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1answer
29 views

How to calculate powers of a permutation in cyclic notation? [on hold]

How do I calculate powers of an 8-cycle (1 2 3 4 5 6 7 8) ?
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0answers
24 views

G acts freely on X. G is paradoxical implies X is also paradoxical

I am proving the Banach-Tarski paradox using a series of small results. For definition of certain terms, see here. Group $G$ acts freely on $X$ i.e. $\operatorname{Stab}(x)=e, \ \forall \ x\in X$. ...
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1answer
34 views

Examples of irreducible representations

Which of the following representations are irreducible? 1) The tautological representation of $D_n$ on $\mathbb{R}^2$ 2) The action of $U(1)$ on $\mathbb{C}$ by multiplication 3) The ...
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2answers
22 views

Group Actions: Verify a Bijective Correspondence

This is an old exam problem: Given an action of $G$ on $X$, we can define $\varphi: G \to S_X$ by the rule $\varphi(g) = \sigma_g$, where $\sigma_g$ is left multiplication by $g \in G$. Prove that ...
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1answer
17 views

Unit group of a field is divisible

In the lecture notes on Valuation theory, in Ex. $1.16$ on page $11$ we are asked to show that: If $k$ is an algebraically closed field, then $k^{\times}$ is a divisible abelian group. Isnt $k = ...
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1answer
24 views

Verify that $A \oplus B$, where $A$ and $B$ are cyclic groups of orders 2 and 3, is the cyclic group of order 6

Let's define $A$ and $B$ as follows: $A$ = {e,a} $B$ = {e,b,2b} Then $A\oplus B= \{\{e+e\},\{e+b\},\{e+2b\},\{a+e\},\{a+b\},\{a+2b\}\}$ which is equal to ...
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1answer
60 views

Aut$(G)\cong \Bbb{Z}_8$

I am looking for a group such that Aut$(G)\cong \Bbb{Z}_8$. Obviously Aut$(\Bbb{Z}_n)\ncong \Bbb{Z}_8$ for any $n$. Also Aut$(D_4)\cong D_4$, neither symmetric/alternating groups are of any help ...
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1answer
13 views

Existence of a special Central cyclic subgroup

I have two related questions. Let $G$ be a finite $p$ group. Can we always choose a central subgroup $N$ of order $p$ not contained in the commutator subgroup? Clearly we cannot do that for extra ...
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1answer
19 views

*$G$-invariant* symmetric bilinear form & $G'=\Bbb Z_2\times\Bbb Z_2$.

I got a problem with the last point I solved all the points, from (a) to (h), but I have no idea how to solve (i): how can I associate a bilinear form to a represtation? What is a $G$-invariant ...
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1answer
20 views

Condition of reversibility of Markov Chain [on hold]

Show that a Markov Chain is time reversible iff $\pi _{i} P_{ij}= \pi _{j} P_{j i}$
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17 views

Tensor product of representations of a product group?

Given some group $G$ that can be written as product of two other groups $$G = G_1 \times G_2 $$ and some representation of this group written in terms of representations of $G_1$ and $G_2$ $$R = ...
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29 views

Is the product of group representations commutative?

Consider, for example, the product of $E_6$ representations $$ 78 \cdot \overline{351}_s \cdot 78 \cdot 351_s, $$ where the $s$ denotes symmetric. Is this equal to $$ 78 \cdot 78 \cdot ...
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5answers
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Why is the commutator defined differently for groups and rings?

The commutator of two elements in a group is defined as $[g, h] = g^{−1}h^{−1}gh.$ In a ring, the commutator of two elements is $[a, b] = ab - ba.$ I'm asking because a ring is a (abelian) group ...
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0answers
36 views

Projective general linear group on discrete valuation ring

Let $R$ be a complete discrete valuation ring and $k$ its residue field. Let $H$ be a finite subgroup of $PGL_2(k)$ such that its order is prime with char($k$). Is there some elementary way to show ...
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Covering groups

I am studying the Steiner system $S(5,6,12)$ and the ternary extended Golay code $\mathscr{C}_{12}$. The automorphism group of the Steiner system is the Mathieu group on twelve elements $M_{12}$ ...
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1answer
28 views

Abelianization of a $p$ group.

Let $G$ be a $d$ generated finite $p$ group. Let $N$ be normal subgroup of order $p$ contained in $[G,G]\cap Z(G)$. Can we say say that $(G/N)/[G/N,G/N]=(G/N)_{ab}$ is a direct product of $d$ cyclic ...
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1answer
58 views

On cyclic decomposition of element in $S_n$

Let $S_n$ be symmetric group and $x\in S_n$ be a permutation of $n$ numbers. Let $|x|=p$, where $n/2<p<n$ is prime. Consider $1^{t_1}2^{t_2}\ldots l^{t_l}$ to be the cyclic decomposition of $x$. ...
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21 views

Proper formulation of one-to-one and onto proofs for group isomorphism

I have to construct an isomorphism for the two groups. I have the isomorphism itself but I'm not sure if my formulation is correct in regard to proving the mapping being 1-1 and onto and I don't want ...
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1answer
84 views

Example of non isomorphic groups with isomorphic group algebras

Below is the construction of two non isomorphic groups, $G_1$ and $G_2$ such that $KG_1 \cong KG_2$ for any field $K$. (My Doubts lie within.) Consider two groups $Q_1=\langle x_1,y_1,z_1\ |\ ...
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1answer
36 views

Conjugates of Sylow $p$-groups in $GL_3(F_p)$

In this list of review questions, there is the following question about $GL_3(F_p)$. Question 1.38. Let $G$ be the group of invertible 3 × 3 matrices over $F_p$, for $p$ prime. What does basic ...
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1answer
115 views

A game from Exercise in Artin's Algebra (Chapter 2 M.13)

I found an interesting problem in Chapter 2 for Artin's Algebra (2nd Ed) in the Miscellaneous section that I haven't been able to figure out. The text of the problem is quoted below. M.13 (a ...
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0answers
61 views

Prove that $|\operatorname{Aut}(G)| \leq \prod_{i=0}^{k}(n-2^i)$ [on hold]

Suppose that $G$ is a group of order $n > 1$, prove that $$|\operatorname{Aut}(G)| \leq \prod_{i=0}^{k}(n-2^i)$$ where $k=\lfloor \log_2 (n-1)\rfloor$.
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1answer
28 views

Derived subgroup of $S_n$ and $D_n$

I know that Derived/Commutator subgroup of $S_3$ is $A_3$ and commutator subgroup of $D_4$ is cyclic of order $2$. But What about derived groups of $S_n$ and $D_n$? How can I calculate them?
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2answers
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Necessary and Sufficient conditions to be a subgroup and/or a normal subgroup.

If $x \in G$, is it possible that $C = \{g^{-1}xg : g \in G \}$ is a subgroup of $G$? Can $C$ be a normal subgroup of $G$? (What are necessary and sufficient conditions to be such a subgroup?) ...
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3answers
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The Stabilizer of the coset for the action of G on $G/H$ by left multiplication.

Let $H$ be a subgroup of $G$. What is the stabilizer of the coset $aH$ for the action of $G$ on $X=G/H$ by left multiplication? So, I think I've done this one correctly: The Stabilizer is of the ...
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22 views

What's the asymptotic of the radius of the Rubik square Cayley graph?

This post is a sequel of The Rubik Square permutation groups, which should be read first to understand the notation. Question: what's the radius$^*$ of the Cayley graph of $G_n$ generated by the red ...
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1answer
38 views

Group Theory: Suggest video lecture (In English)

Please suggest video lecture for following topics in Group Theory. Revision of definition and examples of groups, subgroups. Cyclic Groups, Classification of subgroups of cyclic groups. Permutation ...
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1answer
67 views

The Rubik Square permutation groups

This post was inspired by this webpage of mathematical challenge due to Mickaël Launay (French). Let $G_n$ be the subgroup of $S_{n^2}$ generated by the red arrow permutations as for the following ...
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1answer
263 views

Group $G$ whose center $Z(G)$ is cyclic and with $G/Z(G)$ commutative

I have some issue to solve following exercise. The exercise is from a French book on Algebra (cours d'Algèbre) from Jean Querré. The book is from the 1970's. If the center $Z(G)$ of a group $G$ is ...
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1answer
28 views

Number of Cosets of Intersection of Subgroups

Similar question has been asked on SE before but the problem statement is usually more specific and gives more information (in particular, tells you what to prove), but this problem asks to prove or ...
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1answer
82 views

Is a nontrivial finite group of order $n$ always isomorphic to a subgroup of $GL_{n-1}(\mathbb{Z})$?

I saw this question on an old qualifying exam: Let $G$ be a group of order $n\ge2$. Is such a group always isomorphic to a subgroup of $GL_{n-1}(\mathbb{Z})$? A simpler problem would be to show ...
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4answers
62 views

A problem on order of a Group. [on hold]

Let $G$ be a group of order $8$ and $x$ be an element of $G$ of order $4$. Show that $x^2 \in Z(G)$, the center of $G$. How this result can be proved?
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Proof that the kernel is a normal subgroup of the domain: repeated line

On proofwiki (https://proofwiki.org/wiki/Kernel_is_Normal_Subgroup_of_Domain), the lines corresponding to 'definition of identity' and 'definition of kernel' are identical. Why do we need the second ...