Questions tagged [finite-groups]
Questions on group theory which concern finite groups.
370 questions
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Maximum size of $k$-Sidon set over $\mathbb{F}_2^n.$
Fix $k \in \mathbb{N}$, $k \ge 2.$
Does there exist a subset $A \subset \mathbb{F}_2^n$ such that $|A| \ge c 2^{n/k}$ with some absolutely positive constant $c,$ and satisfying
$$ a_1 + a_2 + \...
user148117
7
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0
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405
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How can I get my hands on McKay's "Finite p-groups" lecture notes?
How can we find Susan McKay's "Finite $p$-groups" lecture notes?
The notes I'm talking about are these.
I emailed Peter Cameron, but he has since moved to a different university, and has no ...
- 4,425
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2
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764
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Proofs of a character identity?
Let $G$ be a finite group, $g \geq 0, k\geq 1 $ integers, and $(C_1,...,C_k)$ a tuple of conjugacy classes of $G$. I am interested in proofs of the following identity
$$
\sum_{(c_1,...,c_k) \in C_1 \...
- 5,349
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2
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780
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Finite groups with a character having very few nonzero values?
A number theorist I know (who studies Galois representations) raised a question recently:
Which finite groups can have an irreducible character of degree at least 2 having only $n=2, 3$, or 4 ...
- 52.9k
7
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1
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Are the distributive permutation groups linearly primitive?
An action of a group $G$ on a set $X \neq \emptyset$ is called transitive if $\forall x,y \in X$, $\exists g \in G$ such that $g.x = y$.
It is called primitive if it is transitive and preserves no non-...
7
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1
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Classification of $p$-groups of order $p^n$ with rank $n-1$
Hello,
i've been looking for a way to classify the non-trivial $p$-groups $G$ that live in an exact sequence of the form
$ 0 \rightarrow \mathbb{Z}/p\mathbb{Z} \rightarrow G \rightarrow (\mathbb{Z}...
- 1,269
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votes
3
answers
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Subgroups of p-groups
If $G$ is a (non-abelian) $p$-group, $|G|=p^n$, $n>3$, then it is elementary that $G$ contains a (normal) abelian subgroup of order $p^2$. It is also true that $G$ necessarily contains a normal ...
- 1,236
6
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1
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Number of points on a linear algebraic group over a finite field
Let $G$ be a linear algebraic group defined over a finite field $\mathbb{F}_q$ as a variety of dimension $d$. What would be a good, simple lower bound for $G(F_q)$?
One can get something fairly nice ...
- 20.2k
6
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2
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Parabolic induction GL(n,Zp)
Let $P$ be a parabolic subgroup of $GL(n)$ with Levi decomposition $P =MN$, where $N$ is the unipotent radical.
Let $\pi$ be an irreducible representation of $M(\mathbf{Z}_p)$ inflated to $P(\mathbf{...
- 11.2k
6
votes
3
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348
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Is there a maximal subgroup of depth 3?
Let's first define what we mean by depth of a subgroup.
Let $G$ be a finite group and $H$ a subgroup. Let $(V_i)_{i \in I}$ and $(W_j)_{j \in J}$ be the irreducible complex representations of $G$ ...
6
votes
1
answer
980
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Group (Co)Homology of Symmetric Group
The question concerns the group homology or group cohomology of symmetric groups.
The entries in groupprops.subwiki.org and in this MO post show the results for the symmetric group S$_4$.
groupprops....
- 10.5k
6
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0
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194
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What is the value of the fourth cohomology class of $\mathrm{Co}_0$ induced by the 24-dimensional representation?
The group $\mathrm{Co}_0$ has a 24-dimensional module. This induces a map $\mathrm H^4(O(24),\mathbb Z) \to \mathrm H^4(\mathrm{Co}_0,\mathbb Z)$. Has this map been computed? Has the right hand side ...
- 54.6k
6
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3
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Restricting the Steinberg representation of $SL_{2n}$ over a finite field to the symplectic group
Let $\text{St}_n(\mathbb{F}_q)$ be the Steinberg module (over $\mathbb{C}$) for $\text{SL}_n(\mathbb{F}_q)$.
What is the irreducible decomposition of the restriction of $\text{St}_{2n}(\mathbb{F}_q)$ ...
- 181
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1
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Doubly primitive groups with simple socle
The classification of doubly transitive groups with simple socle is
known. A good account of such classification can be found for example
in this paper:
Cameron, Peter J. Finite permutation groups ...
- 3,078
6
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0
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430
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More on Moonshine for the Thompson group and weakly holomorphic weight one half modular forms
This question is a follow-up to
Monstrous Moonshine for Thompson group $Th$?
and is based on various comments to that question, in particular S. Carnahan's mention of the
connection to known ...
- 5,546
6
votes
1
answer
678
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Finite Homomorphic images of infinite products of finite solvable groups
I conjecture that:
Every Finite Homomorphic image of an infinite (with arbitrary cardinality) product of finite solvable groups is solvable -- or at least Not a simple (non-abelian) group.
I can ...
- 145
6
votes
1
answer
2k
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Generalization of the fundamental theorem of cyclic groups
Let $G$ be a finite group then the fundamental theorem of cyclic groups can be formulated as follows:
Theorem: $G$ is cyclic iff it admits no two different subgroups with the same order.
proof: see ...
6
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1
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587
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What are the "simplest" polytopes with an automorphism group of $\mathrm M_{11} \hspace {-1.25pt} $?
Do any polytopes have an automorphism group of the smallest of the sporadic groups, the Matthieu group $\mathrm M_{11} \hspace {-1pt} $? Indeed, they must exist. What are the simplest such polytopes ...
- 179
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1
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Automorphisms of powers of finite simple groups
It is a theorem that a finite nontrivial group $G$ has no proper nontrivial characteristic subgroups if and only if $G \cong S^n$ where $S$ is simple and $n > 0$ is the number of copies of $S$ in a ...
- 591
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What is the motivation and purpose of the Floretion group?
When searching through the Oeis, I came across something called a floretion. Based on the context, it seems to be some sort of algebraic structure. I googled it and found nothing that explained their ...
- 1,129
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492
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Centralizer of elements in the upper-triangular matrices
Let $p$ be a prime number and $G=\operatorname{GL}_n ( \mathbb{Z} / p \mathbb{Z}
)$ such that $n\leq p$. Consider the set $U$ of upper-triangular matrices of $G$
having entries of $1$ on the diagonal. ...
- 463
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1
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Are there infinitely many insipid numbers?
A number $n$ is called insipid if the groups having a core-free maximal subgroup of index $n$ are exactly $A_n$ and $S_n$. There is an OEIS enter for these numbers: A102842. There are exactly $486$ ...
6
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1
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262
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Is there some sort of formula for $\tau(S_n)$?
Let $G$ be a finite group. Define $\tau(G)$ as the minimal number, such that $\forall X \subset G$ if $|X| > \tau(G)$, then $XXX = \langle X \rangle$.
Is there some sort of formula for $\tau(S_n)$, ...
- 2,618
6
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1
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567
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Finite simple groups and $ \operatorname{SU}_n $
A follow-up question to Alternating subgroups of $\mathrm{SU}_n $.
$\DeclareMathOperator\PU{PU}\DeclareMathOperator\PSU{PSU}\DeclareMathOperator\PSL{PSL}$Let $ \PU_m $ be the projective unitary group, ...
- 4,081
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Concrete formula for Shapiro's Lemma
I wonder if there is a concrete formula to express the isomorphism in the well known Shapiro's Lemma that $H^i(G, \text{CoInd}_{H}^{G}(M)) \simeq H^i(H, M)$, where $H \subset G$ is a subgroup of $G$, $...
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Which groups have undetectable third U(1)-cohomology?
Let $G$ be a finite group. A categorical Schur detector for $G$ is a set $\mathcal{S}$ of proper subgroups $S \subsetneq G$ such that the total restriction map
$$ \mathrm{rest}_{\mathcal{S}} : \mathrm{...
- 54.6k
5
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2
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309
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A functional equation for a family of functions indexed by the symmetric group $S_3$
$\newcommand{\C}{\mathbb C}$A question asked recently was as follows:
For the symmetric group $G:=S_3$, is it possible to construct functions $t_g\colon\C\to\C$ that satisfy the convolution identity
...
- 128k
5
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Projective representations of a finite abelian group
Projective representations of a group $G$ are classified by the second group cohomology $H^2(G,U(1))$. If $G$ is finite and abelian, it is isomorphic to the direct product of cyclic groups
$$
G\cong ...
- 813
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Does a non-simple perfect group always have a maximal subgroup whose derived subgroup has nontrivial core?
Let $G \neq 1$ be a finite perfect group which is not simple.
Is it true that $G$ necessarily has a maximal subgroup whose derived subgroup
has nontrivial core in $G$?
Remark 1: This holds for all ...
- 347
5
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2
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193
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A question on UCS p-groups
A $p$-group $G$ is called a ${\it UCS}$ $p$-group if $G$ has precisely three characteristic subgroups, namely $1$, $\Phi(G)$ and $G$.
Let $G$ be a finite UCS $p$-group of order $p^{2n}$ such that $\...
- 479
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0
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351
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Adjoint identity on finite nilpotent groups
Let $G$ be a finite nilpotent group. Consider the following (adjoint) identity from [BuPa, Theorem 9.6]:
$$\left(\prod_{\chi \in \mathrm{Irr}(G)} \frac{\chi}{\chi(1)} \right)^2 =\frac{|Z(G)|}{|G|}\...
5
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0
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559
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Atlas of finite groups, Character table of automorphism group of sporadic group
I am consulting ATLAS of finite group for character table of Automorphism Group of sporadic group.
I am reading from Inverse Galois Theory by G. Malle
Let me start with $G=M_{12}$
This(image ...
- 591
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1
answer
343
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Large gaps in Singer planar difference sets?
By a classical result of Singer (1938), for a prime number $p$ the cyclic group $C_n$ of order $n=1+p+p^2$ contains a subset $D$ of cardinality $|D|=1+p$ such that $DD^{-1}=C_n$. Such set $D$ is ...
- 41.9k
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2
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Special automorphisms of extraspecial groups
Let $G$ be an extraspecial group of order $p^{2r+1}$ (where $p$ is an odd prime), and let $V$ be a faithful representation of $G$. Consider the normal subgroup $H$ of $Aut(G)$ consisting of all ...
- 51
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On J.G.Thompson's conjecture about conjugacy classes of finite simple groups
This conjecture in "Unsolved problems in group theory" No.18: 9.24:
Conjecture: every finite simple non-abelian group $G$ can be represented in the form $G=CC$, where $C$ is some conjugacy class of $...
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Burnside group $B(2, 3)$ has $27$ elements, isomorphic to unitringular matrix group $\text{UT}(3, 3)$?
I realized my question here might have been too hard for MSE, so I'm asking it here as well.
The Burnside group $B(d, n)$ is defined as the quotient of the free group on $d$ generators by the normal ...
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Can we classify all finite 2-generated groups $G$ such that if $x,y$ generate $G$, then so does $x,yxy^{-1}$?
Can we classify finite 2-generated groups $G$ satisfying the following property:
For any pair $x,y$ which generate $G$, the pair $x,yxy^{-1}$ also generates $G$.
By the comments, no nontrivial ...
- 10.7k
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Generalized identities of (soluble) groups
Let $G$ be a group. Let us say that $G$ satisfies a generalized identity of degree $n$ if there exist $a_1,a_2,\dots a_n \in G$ such that
$$x^{a_1}x^{a_2}\dots x^{a_n}=1,$$
for all $x\in G$.
Assume ...
5
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0
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299
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A class 3 group of order 243
Let G be a group of order $243=3^5$. We denote by $(G_i)$ its lower central series and assume that $G$ has class $3$ and that $|G:G_2|=|G_3|=9$. We assume moreover that the cubing map factors as a (...
user76083
5
votes
2
answers
375
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a balanced presentation of a cyclic-by-cyclic group?
Let $p>2$ be a prime, $C_p$ be the additive group of integers mod $p$. Then the multiplicative group $\{1,...,p-1\}$ of units in the field $Z/pZ$ is cyclic of order $p-1$, it acts on $C_p$ by left ...
user6976
5
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2
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245
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Counting transitive generators according to coset type
Let $\sigma=(1\;2)(3\;4)\cdots (n-1\; n)$ be a fixed-point-free involution in $S_{2n}$. I want to count permutations $\pi$ such that the group $\langle \pi,\sigma\rangle$ generated by $\pi$ and $\...
- 1,549
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0
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159
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lifting of idempotents in group ring
Let $G$ be a finite group, and let $\pi:G\to Q$ be a surjective group homomophism. The map $\pi:G\to Q$ does not necessarily split, but we can always find a set theoretical splitting $s:Q\to G$. In ...
- 5,039
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745
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Decomposition into irreducible of a representation of the wreath product $S_d\wr S_n$
Let $S_d, S_n$ be the permutation groups of $d,n$ elements.
An intuitive representation of the wreath product $S_d\wr S_n$ is $V_1\otimes...\otimes V_n$, where each $V_i$ is of dimension $d$. Writing ...
- 583
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1
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458
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Does there exist an order in a number field of deg>1 with a map to F_p for all p?
This question is motivated by a computational issue. Suppose $R$ is a product of orders in numberfields such that there is no ring homomorphism $R \to \mathbb Z$, then can one write an algorithm that ...
- 2,224
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0
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159
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Can the numbers of elements of distinct prime orders of a finite simple group coincide? [duplicate]
Does there exist a finite simple group $G$ and distinct prime numbers
$p$ and $q$ dividing the order of $G$ such that the numbers of elements
of $G$ of order $p$ and $q$ are the same?
Remark 1: It ...
- 19.6k
4
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0
answers
228
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Polynomials of growth for finite Heisenberg groups
Take a standard finite Heisenberg group with two standard generators and let's consider its growth polynomial - the polynomial which coefficients are equal to the sphere sizes.
For example for $H_3(Z/...
4
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0
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226
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Possible dimensions for triples of unitary irreducible representations whose tensor product contains the identity
For which triples $\{A,B,C\}$ of positive integers does there exist a (finite or compact) group $G$ with unitary irreducible representations of dimensions $A$,$B$, and $ C$ whose tensor product ...
- 163
4
votes
1
answer
184
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Finite groups of planar homeomorphsims
Let $G$ be a finite subgroup of the group $H$ of orientation-preserving homeomorphisms of the plane that fix the origin. Is $G$ conjugate in $H$ to a group of rotations?
I've been told this result ...
- 726
4
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2
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485
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Transposition Cayley graphs are planar
Consider the Cayley graph $G$ with vertex set the elements of the symmetric group $S_n$ and generating set the set of minimal transposition generators of the group $S_n$, that is the set $S=\{(12),(13)...
- 2,089
4
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2
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393
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Is $(G\rtimes H,H)$ a Gelfand pair iff $G$ is abelian?
Let $G$ be a finite group and $K \subset G$ a subgroup. Then $(G,K)$ is a Gelfand pair if the double coset Hecke algebra $\mathbb{C}(K \backslash G / K)$ is commutative.
Let $H$ be a subgroup of $...