The study of symmetry: groups, subgroups, homomorphisms, and group actions.

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4
votes
1answer
37 views

If $|G|=p^n$, then $p^2 \le |G : G^\prime|$.

Prove that, if $G$ be a p-group of order $p^n$, then $p^2 \le |G : G^\prime|$, where $G^\prime$ is the commutator subgroup of $G$ and $n \ge 2$.
13
votes
1answer
357 views

Why does Group Theory not come in here?

Here is a list of questions that I find quite similar, for the one and only reason that they all show much "symmetry". Which is at the same time my problem, because I don't have a very precise notion ...
2
votes
2answers
30 views

Is that true for given group if their divisors are isomorphic, then their quotient group are isomorphic?

I think there will be counter examples in both cases: For given group $G$ and its normal group $H,K$ (1) If $H$ and $K$ are isomorphic then $G/H$ are $G/K$ are isomorphic (2) If $G/H$ are $G/K$ ...
-2
votes
1answer
47 views

Sylow's theorem and uniqunes of normal supgroup

Let $G$ be a finite group of order $pq,$ where $p$ and $q$ are primes such that $p < q.$ Then how to prove that $G$ has a unique normal subgroup of order $q?$
0
votes
2answers
564 views

Intersection of Normal Subgroups is Normal in Subgroup but Not Group - Fraleigh p. 143 14.35

Show that if H is a subgroup of a group G, and N is a normal subgroup in G, then $H \cap N$ is normal in H. Show by an example that $H \cap N$ need not be normal in G. I can condone the proof hence ...
1
vote
0answers
27 views

How to determine $\left|\operatorname{Aut}(\mathbb{Z}_2\times\mathbb{Z}_4)\right|$

I know that the group $\mathbb{Z}_2\times\mathbb{Z}_4$ has: 1 element of order 1 (AKA the identity) 3 elements of order 2 4 elements of order 4 I'm considering the set of all automorphisms on this ...
1
vote
3answers
40 views

Cayley graph is a tree iff group is free

I am looking at this proof of this claim that the cayley graph is a tree iff g is a free group with generating set S. For the direction '$\implies $' I see that they have assumed that there are two ...
1
vote
0answers
24 views

Subgroups of finite index have finitely many conjugacy classes

is it true the following statement: Let $G$ be a group and let $H$ be a subgroup of $G$. If the index $[G:H]$ of $H$ in $G$ is finite, then $H$ have finitely many conjugacy classes. What I think, is ...
0
votes
2answers
56 views

Is it true that if $\sigma \in S_n,$ then $\sigma^n = \iota$?

I think I remember my abstract algebra professor mentioning in class that if $\sigma$ is any permutation belonging to the symmetric group $S_n,$ then $\sigma^n = \iota,$ the identity permutation. Is ...
-2
votes
0answers
34 views

how many proper subgroups are there in a trivial group{e}? [on hold]

Is there any proper subgroup of a trivial group of order 1? If yes or no how to prove it?
4
votes
1answer
56 views

When is $HK \cong H \times K$?

Suppose $G$ is a group and $H$ and $K$ are subgroups such that $G = HK$ and $H \cap K = \left\{e\right\}$, the identity element of $G$. When can we say that $HK \cong H\times K$? I tried to set up ...
2
votes
0answers
16 views

The relation of determinants between linear transformation.

I am studying the simplicity of PSL and came across the above statement. I don't understand why $\det L_c =c \det L$? (Given two set of basis of the same vectorspace, $v_1,...v_n$ and $w_1,...w_n$, ...
1
vote
2answers
44 views

Let G be an infinite cyclic group. Prove that G cannot have any non-identity elements of finite order.

SO I know that I'm suppose to prove it by contradiction and assume that the element has a positive power. I'm not really sure how to answer it though.
1
vote
1answer
59 views

Does $(\Bbb{Z}/n\Bbb{Z})^\times$ contain $0$?

I recently was told that $(\Bbb{Z}/n\Bbb{Z})^\times$ is cyclic if $n$ is prime. But then this is impossible if it contains $0$. Hence, does $(\Bbb{Z}/n\Bbb{Z})^\times$ contain $0$?
2
votes
0answers
32 views

Every normal subgroup of $GL_n(K)$ either contains $SL_n(K)$ or is contained in $Z$

In this page, it is claimed that if $K$ is a field then every normal subgroup of $GL_n(K)$ either contains $SL_n(K)$ or is contained in $Z$ (the center, which is the scalar matrices). What is the ...
4
votes
3answers
78 views

If $G$ is a finite group and $|G| < |A| + |B|$, then $G=AB$.

Let $G$ be a finite group. Suppose that $A$ and $B$ are to subsets of $G$. If $|G|<|A|+|B|$ prove that $$G=AB.$$
0
votes
0answers
17 views

Subgroups of direct product.

I have problem with finding all subgroups of $\mathbb{Z}_n \times D_m$ and $D_n \times D_m$. First, if $H_1 \leq \mathbb{Z}_n (\text{or} \ D_n)$ and $H_2 \leq D_m$, then $H_1 \times H_2 \leq ...
3
votes
1answer
46 views

Let $G$ be a finite group, $p$ the smallest prime divisor of $|G|$, and $x\in G$ an element of order $p$.

Suppose $h\in G$ is such that $hxh^{−1}=x^{10}$. Show that $p=3$. I am trying to solve this problem using group actions. Let $H$ and $X$ be the subgroups of $G$ generated by the elements $h$ and $x$, ...
2
votes
2answers
41 views

Is this group homomorphism well-defined?

Let $X = \langle a, b \mid aba^{-1}b^{-1} \rangle$ and $Y = \langle a, b \mid aba^{-1}b \rangle$. I want to define $f : X \rightarrow Y$, such that, $f(a) = a$ and $f(b) = b^2$, however I'm having ...
2
votes
0answers
23 views

Natural action of $\operatorname{Aut}(G)$ on sets of subgroups of $G$ of same order is transitive.

I am looking for the classification of those finite groups whose automorphism group acts transitively on sets of subgroups of same order. Let $G$ be a finite group and $d$ be a divisor of the order ...
3
votes
2answers
32 views

How to descent to smaller groups “by chopping off a node of the Dynkin diagram”?

I read in section 2 of this paper : "There is a well-defined chain to descent from $E_8$ to smaller groups by chopping off a node of the Dynkin diagram." What exactly is here referring to ...
3
votes
0answers
136 views
+50

A property of the subgroups lattices

Let $G$ be a finite group. Consider all the subgroups $H$ such that its subgroups lattice $\mathcal{L}(H)$ is distributive (i.e. the group $H$ is cyclic, by Ore's theorem), and among them, let $\{ ...
1
vote
2answers
45 views

Show that $Z(θ^G)≤H$

Suppose $H ≤ G$ and $θ \in Char(H)$. Show that $Z(θ^G)≤H$. ($Z$ is the centre and $θ^G$ is the character induced by $G$)
2
votes
1answer
43 views

If two subsets $S,T\subseteq G$ have sum of cardinalities greater than $|G|$, then $S+T=G$ [duplicate]

Let $S$ and $T$ are two subset of a finite group $(G,+)$ so that $|S|+|T|>|G|$, then Prove that $S+T=G$, where $S+T=\{s+t:s\in S ,t\in T\}$ My effort: It is clear that $S+T\subseteq G$ as ...
4
votes
2answers
99 views

rectangular groups are completely simple and orthodox

Let $S$ be a rectangular group. i.e $S$ is isomorphic to the direct product of a group and a rectangular band.
0
votes
0answers
25 views

Equivalence of right and left cosets of two different subgroups.

Let $A$ and $B$ be two (not necessarily equal) abelian subgroups of $S_5$. If $x$ is an element of $S_5$, under what condition is the following satisfied $$xA = Bx$$ Update: The original question I ...
0
votes
1answer
29 views

The Union of Two Normal Subgroups is also a normal subgroup

I know the statement The Union of Two Normal Subgroups is also a normal subgroup is false. Is there a counter example to show this? I can prove that the intersection is normal, but I can't disprove ...
2
votes
2answers
70 views

Proving that a set is infinite

Consider $G\subset M_2(\mathbb{C})$ where $G =\begin{Bmatrix} \begin{pmatrix} a & 10b\\ b & a \end{pmatrix} & | a,b \in \mathbb{Q},a^2-10b^2=1& \end{Bmatrix} $ . Prove that G is ...
3
votes
1answer
23 views

List the elements and cosets

In the group $\mathbb{Z}_{24}$, let $H=\langle 4 \rangle$ and $N=\langle6\rangle$ a. list the elements in $HN$ (usually write $H+N$ for these additive groups) and $H\cap N$ So I think $H=\langle4 ...
2
votes
4answers
241 views

Problem from Herstein (Group Theory)

This is the problem from Topics in Algebra by I. N. Herstein. Part of Example No. 2.2.9: Let $G$ be the set of all $2 \times 2$ matrices $ \left( {\begin{array}{cc} a & b \\ c & d \\ ...
3
votes
1answer
50 views

Second Isomorphism Theorem

There is one little detail in the proof I would very much like to get your opinion of. Look at where I have circled in red: There it seems that they have used that $\mu_2((hn)N)=h$. But isn't ...
0
votes
1answer
28 views

Normal subgroup corresponding to a relation

Suppose I have a free group on $n$ elements, $FX$, quotient-ed by an element (say, $\langle a, b \rangle/aba^{-1}b^{-1}$), how do I compute the normal subgroup $N$ of $FX$, such that $FX/N$ matches ...
0
votes
1answer
40 views

Finite abelian group - product of its elements [on hold]

Is there a finite abelian group G such that the product of the orders of all its elements is 2^2009?
1
vote
1answer
51 views

Quotient group(Factor group)

Prove that the quotient group $\frac{Z\times Z\times Z}{<(1,1,1)>}$ is an infinite, non-cyclic group. Here Z is the group of integers with operation of addition, $<(1,1,1)$> is the ...
2
votes
1answer
2k views

Drawing subgroup diagram of Dihedral group $D4$

I am reading Fraleigh p. 80 in A First Course in Abstract Algebra, and in the book i see the elements and subgroup diagram of the dihedral group $D_4$. Here are they: Here is how i try to draw ...
1
vote
0answers
17 views

Semidirect product.

I have a problem with representation of this : $(D_n \times D_n) \rtimes \mathbb{Z}_2$, where $\mathbb{Z}_2$ acts on $D_n \times D_n$ by exchanging the two components. $D_n = \langle x, \ y \ | \ x^n ...
1
vote
0answers
18 views

A small cancellation group does not contain $\mathbb{Z}^3$

I read somewhere that a small cancellation group (ie. a group admitting a presentation statisfying the small cancellation condition $C'(1/6)$) does not contain $\mathbb{Z}^3$, but without a precise ...
-2
votes
4answers
57 views

Prove the only homomorphism between groups with coprime orders is trivial. [on hold]

Deduce that if $\gcd(|G|, |H|)=1$ then the only homomorphism $\phi : G \rightarrow H$ is trivial.
0
votes
1answer
59 views

Which group of order 16 is this? [on hold]

$G$ is an abelian group of order $16$ that has elements $a$ and $b$ such that $|a| = |b| = 4$ and $a^2$ does not equal $b^2$. What group is $G$ isomorphic to?
0
votes
1answer
37 views

action on binary tree [on hold]

Let $g$ be an automorphism of Binary tree , say T, then if we restrict $g$ to subtree $T_0$ (means the subtree with root $0$), and denote restricted action by $g|_{T_0}$, then how can I justify that ...
1
vote
1answer
13 views

Why is the rank of a group is equivalent to the maximum number of independent U(1) generators?

I read here http://motls.blogspot.de/2012/04/exceptional-lie-groups.html that the rank of group is "the maximum number of independent U(1) generators". In my understanding the rank of a group ...
5
votes
1answer
38 views

Frobenius kernel is regular normal elementary abelian p-subgroup?

I'm attempting Exercise 3.4.6 in Dixon & Mortimer's book on Permutation Groups: Let $G$ be a finite primitive permutation group with abelian point stabilisers. Show that $G$ has a regular ...
1
vote
0answers
23 views

Spontaneous or not spontaneous symmetry breaking? That is the question.

I have the following system of ODEs: \begin{cases} \frac{du_{i}}{dt}=F_{i}\left(\boldsymbol{u},\boldsymbol{v}\right)+A, & i=1,\ldots M\\ \\ ...
1
vote
1answer
20 views

Mathematical Name for Physical Gauge Symmetries

In physics, when talking about a gauge transformation, we always mean two combined transformations. For example, a $U(1)$ gauge transformation is a combination of $$ \psi \rightarrow e^{ia(x)} \psi ...
1
vote
2answers
50 views

Pushout of a subgroup

Let $G$ be a group, $H\subseteq G$ be a subgroup and $\iota: H\to G$ the inclusion map. What is the "pushout along $\iota$? How is it constructed?
1
vote
1answer
371 views

a conjugate of a glide reflection by any isometry of the plane is again a glide reflection

Q : Prove that a conjugate of a glide reflection by any isometry of the plane is again a glide reflection, and show that the glide vectors have the same length. I know that: the conjugate ...
1
vote
1answer
33 views

Number of $n$-cycles fixed by a permutation of $S_n$.

Let $\sigma\in S_n$ be a permutation consisting of $m$ $d$-cycles. I want to show that the number of $n$-cycles $\sigma$ commutes with is $\phi(d)m^{d-1}(d-1)!$. If I write $\sigma=(a_{1, 1},\ldots, ...
0
votes
0answers
35 views

$Z(G)$ acts on set of conjugacy classes by left multiplication

Let $z\in Z(G)$ then one can say that $(zx)^g=zx^g$. But it means that multiplication by $z$ create a bijection from conjugacy classes of $x$ to conjugacy classes of $xz$. Let ...
1
vote
2answers
26 views

What is the $l$ in this group? [on hold]

What is the $l$ in this group? $a^7=e$, $b^3=e$, $b^{-1}ab=a^2$, $ba=a^lb$. $l=?$
0
votes
1answer
31 views

Show that the alternating group $A_9$ has no subgroups of index 8?

So far, I believe it's a proof by contradiction. Suppose that $H \leq A_9$ with $[A_9 : H] = 8$.. $|H| = |A_9|*8$(which is a large number)? then would this involve the 3-cycles? Quite stumped. Thank ...