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.

learn more… | top users | synonyms (2)

0
votes
1answer
27 views

Generating set of a group and even subgroup.

Let $G=\langle S\rangle$. Now let $H=\{x_1x_2\cdots x_m\mid x_i\in S \cup S^{-1},\ i\le m\in\Bbb N,\ m\text{ is even}\}$ I'm trying to prove that $[G:H]=1$ or $2$. I started doing this by proving ...
1
vote
0answers
39 views

Why is $M_{21}\cong PSL_3(\mathbb{F}_4)$?

I have recently been reading about the Large Mathieu Groups in "The Finite Simple Groups" by Robert Wilson and I have not come across any explanation of this isomorphism as of yet. In this context, ...
1
vote
0answers
31 views

Prove that if $\tau \in N_{S_A}(H)$ then $\tau$ stabilizes the sets $F(H)$ and $A \backslash F(H)$

Prove that if $\tau \in N_{S_A}(H)$ then $\tau$ stabilizes the sets $F(H)$ and $A \backslash F(H)$ $H$ is the set of fixed points on $A$ $A$ : set, $H \le S_A$, $F(H) = \{ a \in A : \sigma (a) = a, ...
1
vote
1answer
21 views

Prove the set of permuations which permute only finitely many elements is a normal subgroup

Let $A$ be a non-empty set and let $X$ be a subset of $S_A$ Now let $F(X) = \{a \in A : \sigma(a) = a, \forall \sigma \in X\}$, $M(X) = A\backslash F(X)$, and $D = \{ \sigma \in S_A : \mid ...
2
votes
2answers
45 views

In a Group, is the existence of the left identity equivalent to the existence of the unique two sided identity?

I've read many definitions in different books, and some of them specifically point out in their definition the existence of left inverse, left identity, and associativity. But the grand majority does ...
6
votes
3answers
58 views

$H$ be a proper subgroup of finite group $G$ such that $H \cap gHg^{-1}=\{e\} , \forall g \in G \setminus H$ , then $|\cup gHg^{-1}|>\dfrac 12 |G|+1$

Let $H$ be a proper subgroup of finite group $G$ such that $H \cap gHg^{-1}=\{e\}$ for all $g \in G \setminus H$. Then is it true that $$|\cup_{g \in G \setminus H}gHg^{-1}|>\dfrac 12 |G|+1$$ If ...
2
votes
0answers
31 views

Finite Groups With Exactly Two Automorphisms [duplicate]

Is there an easy way how to characterize the groups with exactly two automorphisms? I was able to find the following finite groups with exactly two automorphisms: $\mathbb{Z}_3$, $\mathbb{Z}_4$, ...
2
votes
1answer
34 views

Groups With Exactly One Maximal Subgroup

I understand that when $G$ has exactly one maximal subgroup (inclusion-wise), then $G$ has to be cyclic. But is it possible to determine all possible groups with exactly one maximal subgroup?
3
votes
1answer
47 views

How can I check whether the group $[16,13]$ in GAP with $3$ generators can be generated by $2$ elements?

The group $[16,13]$ in GAP has structure $(C4\times C2):C2$ and is generated by the permutations $(1234)(5678)$ , $(15)(26)(37)(48)$ and $(57)(68)$ . The group $[16,3]$ in contrast with the same ...
3
votes
1answer
46 views

On groups with presentations $ \langle a,b,c|a^2=b^2=c^2=(ab)^p=(bc)^q=(ca)^r=(abc)^s=1\rangle $…

$$ \langle a,b,c|a^2=b^2=c^2=(ab)^p=(bc)^q=(ca)^r=1\rangle =\Delta(p,q,r) $$ This is a presentation of a triangle group $\Delta(p,q,r)$, a special kind of Coxeter group. What about the following ...
3
votes
1answer
51 views

Which non-abelian finite groups have the property that every subgroup is normal?

If $G$ is an abelian group, every subgroup $H$ of $G$ is normal. I searched for non-abelian finite groups $G$ , such that every subgroup is normal and GAP showed only the groups $G'\times Q_8$ , ...
0
votes
0answers
21 views

G-set of 8 colors and 6 sides

"How many distinguishable wooden cubes can be painted if u use 8 colors (different colors on every side)" I have solved this question using Burnside's lemma ...
0
votes
1answer
47 views

Prove that (01) and (01…n-1) generate Sn? [duplicate]

Show that every element of Sn can be written as an arbitrary product of the elements (01) and (01...n-1). I understand that this can be solved using induction, and I've set up my base cases. ...
0
votes
2answers
47 views

Two ways to decompose $\mathbb Q[G]$ for $G = \{1,g,g^2\}$ and their interrelation.

Let $G = \{1,g,g^2\}$ be a cyclic group of order $3$. Consider the group ring $\mathbb Q[G]$. Then we have the two $G$-invariant simple subspaces $$ I = \{ a_0 + a_1 g + a_2 g^2 : a_0 = a_1 = a_2 \} ...
-1
votes
2answers
87 views

Only two groups of order $10$: $C_{10}$ and $D_{10}$

Show that up to isomorphism there exist only two groups of order 10: $C_{10}$ and $D_{10}$. I need some help on this question. I only know the basic definitions of isomorphism. Any hints?
0
votes
2answers
26 views

“conjugate to/with” or “conjugated to/with”, a terminology question in group theory.

This is a terminology question from a non-native English speaker. Let $G$ be a group and $a,b\in G$ such that there exists $c\in G$ verifying : $$b=cac^{-1} $$ I could say : the element $a$ is ...
0
votes
0answers
22 views

Inverse element in semidirect product

If K and Q are both groups and $h:Q\rightarrow \text{Aut}(K) $ is a homomorphism then the group operation for the semidirect product $K\rtimes_hQ$ is: $$(k_1,q_1)*(k_2,q_2)=(k_1h(q_1)(k_2),q_1q_2)$$ ...
1
vote
0answers
21 views

How many homomorphisms are there $t: \mathbb{Z}_3 \times U_8 \to S_5$?

This was a question in our quiz today and no one in class knew how to answer it correctly or are not sure). How many homomorphisms are there $t: \mathbb{Z}_3 \times U_8 \to S_5$, where $U_8$ is ...
0
votes
1answer
31 views

Isomorphic to Subgroup of even permutations

True or False Every finite group of odd order is isomorphic to a subgroup of $An$, the group of all even permutations. The question was in entrance exam. I think there is counter example to this ...
0
votes
1answer
29 views

Normal Subgroups and Properties

Suppose we have the normal subgroups $H,J\subset G $ with the property $|G|=|H|\cdot |J|$ and $H\cap J={e}. $ $ $Prove that $H\times J\cong G$. I don't really know how to approach this one. I ...
0
votes
0answers
36 views

How can I prove that $G$ is not a simple group. [duplicate]

I only have that $|G|=4n+2$ for $n \in \mathbb{N}$. And I have to prove that G is not a simple group.
0
votes
1answer
27 views

The conjugation Group action is continuos

How can I prove that the group action from $G\times G\to G$ defined by $(g,x)\mapsto gxg^{-1}$ is a continuos function? I tried to use the known facts that multiplication and $(x,y)\mapsto xy^{-1}$ ...
0
votes
0answers
23 views

How to prove that a matrix lie group is (or is not) simple

I am having some difficulties in proving that some matrix lie groups are or are not simple. I am wondering what I should ask myself to solve this problem quickly. Should I look for normal subgroups? ...
-3
votes
0answers
35 views

$S_X = \{f(x) = x : \text{ bijective} \}$. prove $S_x$ is isomorphic to $S_n$

Aright, to start $S_n$ is the Symmetric group and $S_X = \{x_1, x_2, \ldots x_n\}$. Going through the mapping $\phi(S_X) \to S_n$, I'm not sure how I'd show this mapping and the first thought that ...
1
vote
2answers
70 views

What's the name for the property for which $x + x = 0 \Longleftrightarrow x = 0$?

I have a set $\mathbb{S}$ for which I have defined an operation: addition ($+ : \mathbb{S} \times \mathbb{S} \rightarrow \mathbb{S}$). The structure $(\mathbb{S}, +)$ is a group. I have shown that if ...
0
votes
2answers
52 views

If $G \cong \mathbb Z/3\mathbb Z$, then $\mathbb R[G] \cong \mathbb R \times \mathbb C$

Let $G = \{1,g,g^2\}$ be the cyclic group of order three. Consider the group ring $\mathbb R[G]$, then $\mathbb R[G] \cong \mathbb R \times \mathbb C$ with the isomorphism $$ \varphi(1) = (1,0), ...
1
vote
2answers
35 views

$G$ be a group of order $pn$ , where $p$ is a prime and $p>n$ , then is it true that any subgroup of order $p$ is normal in $G$?

Let $G$ be a group of order $pn$ , where $p$ is a prime and $p>n$ , then is it true that any subgroup of order $p$ is normal in $G$ ? ( I know that any subgroup of index smallest prime dividing ...
2
votes
0answers
22 views

Topological/geometric interpretation of conjugacy separability

We say that a group $G$ is residually finite if for every non-trivial $g \in G$ there is a a finite group $Q$ and a surjective homomorphism $\phi \colon G \to Q$ such that $\phi(g) \neq_Q 1$. This ...
1
vote
2answers
57 views

Does there exist a surjective homomorphism from $(\mathbb R,+)$ to $(\mathbb Q,+)$ ?

Does there exist a surjective homomorphism from $(\mathbb R,+)$ to $(\mathbb Q,+)$ ? ( I know that there 'is' a 'surjection' , but I don't know whether any surjective homomrophism from $\mathbb R$ ...
0
votes
0answers
31 views

Quotient of $S_4$ by a normal subgroup

Is this true? > Quotient of $S_4$ by a normal subgroup of order 4 is Abelian ? As the group is of order 4!/4=6, so it is either $S_3$ or $\mathbb Z_6$. how to decide? please help.
0
votes
0answers
30 views

Simple non-abelian subgroup

How can I show that If H is a simple and non-abelian subgroup of group G then $ H< [G,G] $.THank you for all your answers.
0
votes
0answers
20 views

Image of Hall subgroup is a Hall subgroup

Let $\pi$ be any set of prime numbers. $H \subset G$ is a Hall $\pi$-subgroup if no pime dividing the index $|G:H|$ lies in $\pi$ and every prime divisor of $|H|$ lies in $\pi$. Let $\varphi: G ...
0
votes
0answers
25 views

freeness of vector spaces and abelian groups

This question is continuation of my previous question Extension of vector spaces and abelian groups Given a diagram of linear transformations of $K$ vector spaces $$B\xrightarrow{\epsilon} ...
0
votes
1answer
32 views

Prove that, considering $\mathbb{R}$ as an additive group, we have $SO_2(\mathbb{R})\cong \mathbb{R}/2\pi\mathbb{R}$ .

Let $SO_2(\mathbb{R})$ be the group of rotations of the circle under the operation of composition. Prove that, considering $\mathbb{R}$ as an additive group, we have $SO_2(\mathbb{R})\cong ...
4
votes
1answer
58 views

Alternating group on infinite sets

It is well known that the only normal subgroup of $S_n$ is $A_n$ when $n\geqslant 5$, and that $A_n$ is also simple. Furthermore, $A_{\infty}$, the even permutations on $\mathbb{N}$, is also simple. ...
0
votes
0answers
16 views

Why is $(A \otimes_\mathbb{Z} V_p )\bigcap R(G) = V_p$

I am reading the proof of brauers theorem in Serres book Linear Representations of finite groups and I have trouble understanding Lemma 5(page 75). Let $G$ be a group of order $g$. First let $g=p^nl$ ...
1
vote
1answer
58 views

Where does the group $\mathbb Z/(a)\oplus \mathbb Z/(a^2)\oplus \cdots $ arise?

Let $a>1$ be an integer, and consider the infinite abelian group $$ V_a=\bigoplus_{j=1}^{\infty}\mathbb Z/{a^j\mathbb Z}. $$ Can anyone provide references to places where this (or related) groups ...
2
votes
1answer
45 views

Is the definition given by the GAP-manual equivalent to the one given in the site? [on hold]

Here http://groupprops.subwiki.org/wiki/Normalizer_of_a_subset_of_a_group the normalizer of a subset of a group is defined. GAP gives the following description of the Function IsNormal : 39.3.6 ...
-1
votes
0answers
23 views

A normal subgroup of a finite $p$-group is in the center

I am trying to prove the following (it is an exercise from Rotman): Let $G$ be a finite $p$-group and $N$ its normal subgroup of order $p$ ($p$ is a prime number). Then $N$ is a subgroup of the ...
3
votes
0answers
22 views

oligomorphic subgroups of $S_\infty$

Is it true that every oligomorphic subgroup of $S_\infty$ is not abelian? A subgroup of $S_\infty$ is said oligomorphic if its action on $\mathbb N^n$ has only finitely many orbits for each ...
0
votes
1answer
17 views

How can we prove that the group order is an upper bound both for the number and the dimensionalities of the irreducible representations?

For example, in $S_{3}$ there is 6 number of members for the group while there are 3 different irreducible representation with dimensinalities of 1, 1 and 2 in which gives: $$\sum_{\mu}{n_{\mu}^2} = ...
0
votes
0answers
34 views

about a finite by nilpotent group

I was reading the proof of the following lemma let $G=XM$ xhere $M$ is a normal divisible abelian subgroup and $X$ is a torsion subgroup of $G$.If $G$ is a finite by nilpotent group then ...
0
votes
0answers
30 views

$G_\delta$ subgroups of a Polish Group

Let $X$ be a Polish Group. It's known that every its Polish subgroup is a $G_\delta$. Pick one of them, say $V$. Is it true that $V$ is the intersection of open subgroups? Thank you
0
votes
1answer
37 views

Polish subgroups of $S_\infty$

Let $S_\infty$ considered as Polish Group. Prove that every Polish subgroup of $S_\infty$ has the following form: $\overline{{\left \langle X \right \rangle}}$, where $X$ is a countable subset of ...
1
vote
1answer
39 views

If $G = G_{\alpha} \rtimes R$ and $R$ is non-abelian of order $27$ and exponent $3$, then $G_{\alpha} \cong C_2\times C_2, D_3$ or $D_4$

Suppose that $G$ is a transitive permutation group and let $R$ be a regular normal subgroup which is isomorphic to the non-abelian group of order $27$ and exponent $3$. Then $G = R \rtimes ...
2
votes
2answers
33 views

Extension of vector spaces and abelian groups

While reading about modules from Hilton & Stammbach's Homological algebra, I saw the following statement : $\Lambda$ is a ring. $\Lambda$ modules are generalizations of vector spaces and ...
2
votes
0answers
18 views

Factorization of permutations into two factors with fixed number of cycles, plus a placement condition

I have asked this question in MathOverflow, but it received no answers, so I am posting it here. In my recent work I have been led to consider the following type of permutation factorizations. Let ...
-2
votes
0answers
27 views

Proof that $\exp(a) \cdot r \cdot \exp(-a) =\ exp(ad_a) \cdot r$ [closed]

Let a nilpotent element of associative algebra over field with zero characteristic and $ad_a: b \to [a, b] = ab -ba,$ $b \in R $. Proof that $\exp(a) r \exp(-a) = \exp(ad_a)r (r \in R) $
0
votes
0answers
33 views

Origin of the name “Cauchy sequence” in abstract algebra

I know the origin in analysis as it is derived from our good old chap Cauchy himself. However I have read about Cauchy sequences in algebra, I'll use groups for this one, let $G$ be a group and ...
0
votes
1answer
44 views

The number of elements in the special linear group over the finite field $\mathbb{Z}/p$ [closed]

I have $SL_{2}\{\mathbb{Z}/p\}$ for $p$ prime and $\mathbb{Z}$ integers. How do I show that this is a subgroup of $GL_{2}\{\mathbb{Z}/p\}$ and find the number of elements in $SL_{2}\{\mathbb{Z}/p\}$?