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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Maximal closed subgroups in algebraic groups

Let $G \leq GL(V)$ be an affine algebraic group, over an algebraically closed field. Say that $M$ is a proper subgroup of $G$ which is maximal among the closed proper subgroups of $G$. Does $M$ have ...
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1answer
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Determine the isomorphism class of $\mathbb Z^3 / M$ for the subgroup $M$ of $\mathbb Z^3$generated by $(13,9,2),(29,21,5),(2,2,2)$

The problem seems not so hard. My confusion rise from the statement in the solution above that "This question is equivalent to reducing the matrix via row and column operations". Please see the ...
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23 views

solution verification: find center of this group

we have a group $H$ = $\begin{cases} A= \begin{bmatrix}1 & a & b\\ 0 & 1 & c\\0 & 0& 1 \end{bmatrix} : a,b,c \in \mathbb{Q} \end{cases}$ then find its center $ Z(H)$. My ...
3
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3answers
61 views

Which of the following statements are correct?

T he order of the smallest possible non trivial group containing elements $x$ and y such that $x^7=y^2=e$ and $yx=x^4y$ is (A) $1\space\space$ (B) $2\space\space$ (C) $7\space\space$ (D) $14$
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how to generate equivalence relations for quotient group and verify them and what input relations is suitable for subgroup

1. similar gap system, which function can output coset table ? if find example below, ...
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Generators of the group of invertible elements of the ring $\mathbb{Z}_{14}$—are they multiplicative or additive?

When I was asked to find the generators of the group of invertible elements of the ring $\mathbb{Z}_{14}$, which are denoted as $\phi(14)$, I did not realize whether the generators are multiplicative ...
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3answers
59 views

Looking for help to understand example of Group

I am looking for someone to help me to understand what is going on in the following example, from Hersteins "Topics in Algebra". It says, Let $G$ be the set of all $2*2$ matrices $$\begin{pmatrix} ...
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1answer
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If $G$ is a group of order $2^nm$, where $m$ is odd and $(m-1)!<2^n$, show that $G$ is not simple.

If $G$ is a group of order $2^nm$, where $m$ is odd and $(m-1)!<2^n$, show that $G$ is not simple. I started out by trying to prove this using the Sylow theorem, but it led nowhere. I was able to ...
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2answers
33 views

From group isomorphisms to algebra isomorphisms

Let $A$ be an algebra and let $A^{\ast}$ be the subset of units (that is, invertible elements) of $A$. Then $A^{\ast}$ is a group under the multiplication of $A$. Let $f^{\ast}:A^{\ast}\to A^{\ast}$ ...
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2answers
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Isomorphism between $H/H\cap N$ and $HN/N$

Suppose we have a group G under operation +, and let H be a subgroup and N a normal subgroup. I want to prove that $H/H\cap N$ is isomorphic to $HN/N$. Where, if I am not mistaken: $H/H\cap N = ...
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definition of line complex in projective space

In a paper of "R.H.Dye", which you can find here: http://link.springer.com/article/10.1007%2FBF02413785#page-1, I face with a mathematical object "line complex in projective space PG(2n−1,q), I need ...
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1answer
14 views

Explicitly compute the trace for the tautological representation of $D_4$ of $\mathbb{R}^2$.

Fix a finite dimensional representation $\rho: G \longrightarrow GL(V)$ of $G$. Its trace is defined as the function $tr:G \longrightarrow F$ defined by $tr(g) = tr(\rho(g))$. Explicitly compute ...
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3answers
44 views

Conjugacy Classes of a group G - Intuitive Understanding

How can I intuitively understand conjugacy classes of a group G. I feel I have a strong understanding of Equivalence Relations, and just completed the proof showing that conjugacy is an equivalence ...
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0answers
50 views

Upper bound on groups of order 60

I am aware of the fact that there are 13 non-isomorphic groups of order 60 but the proof of this is really long and something that I cannot present in a few minutes. Hence, I need to give a short ...
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4answers
77 views

In need of a group theory textbook.

I am in need of a group theory textbook for a good summer review. I have already studied from various books (mostly "group theory" part from basic algebra books) and the lecture notes of my teacher, ...
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37 views

Group isomorphism exercise

I am trying to solve the following problem: Let $G$ be a finite group such that there exists an isomorphism $\phi:f:G/Z(G) \to \mathbb Z_3 \oplus \mathbb Z_3$ and $x \in G$ such that ...
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0answers
31 views

Why is the trivial group a zero object for the category of groups, but the empty set isn't a zero object for the category of sets? [duplicate]

I understand that the zero ring can't be a zero object for the category of rings, because in that case the 'arrows' are ring homomorphisms which, by definition, but maintain the unit. But in the ...
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29 views

A qualifying-exam problem related to Clifford theorem in representation theory

I am not sure : does this problem can be solve directly by Maschke's theorem, which states that every representation of a finite group is completely reducible. Perhaps I made some stupid ...
3
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1answer
55 views

Exercise from Serre's “Trees” - prove that a given group is trivial

In Serre's book "Trees" on page 10 the following exercise is given: Show that the group defined by the presentation $$x_2x_1x_2^{-1}=x_1^2, \hspace{7pt} x_3x_2x_3^{-1}=x_2^2, \hspace{7pt} ...
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How do roots act on weights?

In Lie theory it's possible to compute things very explicit using tensor methods. For example, we can use an explicit matrix for each generator $T^a$ and compute the "action" of this generator on an ...
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1answer
31 views

Why can we assume $N$ to be a $p$- group?

Let $G$ be a finite solvable group such that if three distinct primes $p,q$ and $r$ divides $|G|$ then $G$ does not contain any element of order the product of two primes and $G$ is minimal w.r.t ...
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1answer
45 views

Are there any examples of non-abelian subgroups of abelian group? [on hold]

Are there any examples of non-abelian subgroups of abelian group?
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79 views

Group as a $\mathbb Q$-vector space

Let $G$ be a torsion free abelian group of having $n$ number of maximally rationally independent elements $r_{1}, r_{2}, ..., r_{n}$ and assume that $G$ is not finitely generated. Is this correct ...
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1answer
42 views

Group of order $2m$ where $m$ odd has a subgroup of index 2. [duplicate]

Show that a group $G$ of order $2m$, where $m$ is odd, has a subgroup of index $2$. I am feeling a little dubious about my proof. Let $G$ act on itself by left multiplication to induce the ...
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1answer
26 views

Generating Finite Groups By Random Premultiplication With Generators

Let $G$ be a finite group with identity $e$ and $S$ be a set which generates $G$. Is it always possible to define a procedure of the form: Start with $x=e$. With probability $p_1$, replace $x$ with ...
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1answer
51 views

Looking for various proofs there is no faithful representation $\rho:S_4\rightarrow GL(\mathbb{R}^2)$

I stumbled on the question when thinking about a representation $\rho:D_4\rightarrow GL(\mathbb{R}^2)$ as symmetries of the four points $\{(\pm 1, \pm 1)\}$. There didn't seem to be a good geometric ...
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3answers
67 views

Find all of the homomorphisms $ \varphi: S_3 \to \mathbb{Z}_{4} $.

I'd like to find all of the homomorphisms $ \varphi: S_3 \to \mathbb{Z}_{4} $. What I've tried so far: I tried to do $ \varphi(Id) = \bar{0} = \bar{4} $ (as somebody used here). But then I realized ...
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2answers
44 views

Finitely generated abelian group with certain properties

Problem Characterize all finitely generated abelian $G$ such that every proper subgroup of $G$ is cyclic, $G$ contains exactly two proper subgroups, and for each pair of subgroups $S$,$T$ in $G$ ...
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Groups of the from $gMg$ in a monoid where $g$ is an idempotent

Let $(M, \cdot)$ be a finite monoid with identity $e$. It is easy to see that $gMg = \{ gxg : x \in M \}$ forms a monoid with identity $geg = g$ if $g$ is an idempotent. If $gMg$ contains no ...
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1answer
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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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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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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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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 ...
2
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1answer
58 views

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

I was looking for a proper subgroup of $\mathbb Q$ which is not finitely generated under the addition operation. We know every finitely generated subgroup of $\mathbb Q$ is cyclic. For a proper ...
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1answer
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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
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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
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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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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
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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
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Let $G,*$ a group and $a,b,c,d \in G$. Prove that … [on hold]

Let $\langle G,*\rangle$ be 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.
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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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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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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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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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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
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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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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
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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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24 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
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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 = ...