What is Subgroup: Definition and 290 Discussions

In group theory, a branch of mathematics, given a group G under a binary operation ∗, a subset H of G is called a subgroup of G if H also forms a group under the operation ∗. More precisely, H is a subgroup of G if the restriction of ∗ to H × H is a group operation on H. This is usually denoted H ≤ G, read as "H is a subgroup of G".
The trivial subgroup of any group is the subgroup {e} consisting of just the identity element.
A proper subgroup of a group G is a subgroup H which is a proper subset of G (that is, H ≠ G). This is usually represented notationally by H < G, read as "H is a proper subgroup of G". Some authors also exclude the trivial group from being proper (that is, H ≠ {e}).If H is a subgroup of G, then G is sometimes called an overgroup of H.
The same definitions apply more generally when G is an arbitrary semigroup, but this article will only deal with subgroups of groups. The group G is sometimes denoted by the ordered pair (G, ∗), usually to emphasize the operation ∗ when G carries multiple algebraic or other structures.

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  1. A

    Subnormal p-Sylow Subgroup of Finite Group

    I am self-studying a class note on finite group and come across a problem like this: PROBLEM: Let ##G## be a dihedral group of order 30. Determine ##O_2(G),O_3(G),O_5(G), E(G),F(G)## and ##R(G).## Where ##O_p(G)## is the subgroup generated by all subnormal p-subgroups of ##G##; ##E(G)## is the...
  2. H

    Transitive subgroup of the symmetric group

    Hi, I need help in proving the following statement: An abelian,transitive subgroup of the symmetric group Sn is cyclic,generated by an n-cycle. Thank's in advance
  3. T

    Quick question about subgroups of "odd" dihedral groups

    Will odd dihedral groups (e.g. d6, d10, d14) only have the identity, σ, τ and itself as subgroups as any power of σ with τ generates the entire group? If so would the subgroup lattice of d14 just be: d14 → σ & τ → e Thanks!
  4. K

    SU(3) defining representation (3) decomposition under SU(2) x U(1) subgroup.

    I have been reading Georgi "Lie Algebras in Particle Physics" and on page 183 he mentions how that the SU(3) defining representation decomposes into an SU(2) doublet with hyperchage (1/3) and singlet with hypercharge (-2/3). I am confused on how he knows this. I apologize if this is not the...
  5. M

    SO(N) adjoint rep. under SO(3) subgroup

    Hi. I'm having trouble figuring out how SO(N) adjoint rep. transforms under a SO(3) subgroup. Unlike SU(N), SO(N) fundamental N gives \begin{equation} N \otimes N = 1 \oplus A \oplus S \end{equation} So the \begin{equation} S \end{equation} part really bothers. Can you give a help?
  6. TrickyDicky

    SU(2) as a normal subgroup of SL(2, C)

    SU(2) matrices act isometrically on the Riemann sphere with the chordal metric. At the same time the group of automorphisms of the Riemann sphere is isomorphic to the group SL(2, C) of isometries of H 3(hyperbolic space) i.e. every orientation-preserving isometry of H 3 gives rise to a Möbius...
  7. PsychonautQQ

    Modern Algebra unified subgroup question

    Homework Statement If H and K are subgroups of G, show HUK is a subgroup of G if and only if H < K or K < H ( the < meaning that all the elements of H are in K or all the elements of K are in H). Homework Equations None The Attempt at a Solution I believe the problem here is HUK might...
  8. 1

    Subgroup proof - is this even true?

    Homework Statement Prove that G cannot have a subgroup H with |H| = n - 1, where n = |G| > 2. Homework Equations The Attempt at a Solution Counter-example, the multiplicative group R and its subgroup, multiplicative group R+. Or, the additive group Z, and its subgroup of integer...
  9. J

    Proving N is a Normal Subgroup of G

    Homework Statement N is a normal subgroup of G if aNa^-1 is a subset of N for all elements a contained in G. Prove that in that case aNa^-1 = N. Homework Equations The Attempt at a Solution Given: N is a normal subgroup of G if aNa^-1 is a subset of N for all elements a contained...
  10. J

    Center of Factor Group Is Trivial Subgroup

    Homework Statement Prove that the center of the factor group G/Z(G) is the trivial subgroup ({e}). Homework Equations Z(G) = {elements a in G|ax=xa for all elements x in G} The Attempt at a Solution I need to prove G is abelian, because G/Z(G) is cyclic, right? Then I can say that...
  11. Greg Bernhardt

    What is a Subgroup? Definition, Equations & Explanation

    Definition/Summary A subgroup H of a group G is a set of elements of G with G's group operation where H is also a group. The identity of G is also in H. The identity group and G itself are both trivial subgroups of G. With a subgroup, one can partition a group's elements into left cosets...
  12. Greg Bernhardt

    What is a Commutator Subgroup?

    Definition/Summary The commutator subgroup of a group is the subgroup generated by commutators of all the elements. For group G, it is [G,G]. Its quotient group is the maximal abelian quotient group of G. Equations The commutator of group elements g, h: [g,h] = g h g^{-1} h^{-1} The...
  13. M

    MHB The subgroup of S_4 <σ,τ> is the whole S_4

    Hey! :o I am looking at the following exercise: $$\sigma=(1 \ \ \ 2 \ \ \ 3), \ \ \ \tau=(1 \ \ \ 4) \ \ \ \in S_4$$ Calculate the following permutations and notice that they are different from each other and also different from $\sigma, \tau, id$. Show that the subgroup of $S_4$ that is...
  14. D

    MHB Abstract Algebra Sylow Subgroup

    I have a question about abstract algebra so if someone could help me answering this question please ... Suppose P,P' are 3-Sylow subgroup, and let Q be their intersection and N the normalizer of Q. Problem: Explain why is the order of N divisible by 9 ? Thanks for your help. Regards,
  15. X

    Preimage of arbitrary subgroup.

    Suppose θ: A → B is a homomorphism. And assume S ≤ B. Is it necesarily true that if S is a subgroup, that is not completely contained in the range, its preimage forms a subgroup?
  16. W

    Embedding Group as a Normal Subgroup

    Hi, let G be any group . Is there a way of embedding G in some other group H so that G is normal in H, _other_ than by using the embedding: G -->G x G' , for some group G'? I assume this is easier if G is Abelian and is embedded in an Abelian group. Is there a way of doing this in...
  17. O

    MHB What are Cyclic Subgroup Generators and How Do We Determine Them?

    i am having a difficulity understanding the concept of cyclic subgroup generators. may I be given an explanation with examples if possible of how you determine whether a function is a subgroup and when they say list all cyclic subgroups eg <Z_10,+>. show that Z_10 is generated by 2 and 5
  18. J

    Exploring Subgroup Inverse Maps in Group Theory

    Homework Statement For a group G consider the map i:G\rightarrow G , i(g)=g^{-1} For a subgroup H\subset G show that i(gH)=Hg^{-1} and i(Hg)=g^{-1}H Homework Equations The Attempt at a Solution I know that for g_1,g_2 \in G we have i(g_1g_2)=(g_1g_2)^{-1}=g_2^{-1}g_1^{-1} Then...
  19. S

    Subgroup of a Quotient is a Quotient of a Subgroup

    Homework Statement I'm trying to prove the statement "Show that a subgroup of a quotient of G is also a quotient of a subgroup of G." Homework Equations See below. The Attempt at a Solution Let G be a group and N be a normal subgroup of G. Let H be a subgroup of the quotient...
  20. D

    Commutator subgroup a subgroup of any Abelian quotient group?

    I am new to group theory, and read about a "universal property of abelianization" as follows: let G be a group and let's denote the abelianization of G as Gab (note, recall the abelianization of G is the quotient G/[G,G] where [G,G] denotes the commutator subgroup). Now, suppose we have a...
  21. T

    Find Commutator Subgroup of Frobenius Grp of Order 20: Defn Explained

    1) Find the commutator subgroup of the Frobenius group of order 20. 2) I have the Cayley table. 3) What is the definition of a commutator subgroup? I am absolutely sure we haven't heard this term all semester.
  22. Advent

    MHB Closure of a Subgroup of GL(2,C)

    Let GL(2;\mathbb{C}) be the complex 2x2 invertible matrices group. Let a be an irrational number and G be the following subgroup G=\Big\{ \begin{pmatrix}e^{it} & 0 \\ 0 & e^{iat} \end{pmatrix} \Big| t \in \mathbb{R} \Big\} I have to show that the closure of the set G is\bar{G}=\Big\{...
  23. G

    Using generators to check for a normal subgroup

    Homework Statement Perhaps I should say first that this question stems from an attempt to show that in the group \langle x,y|x^7=y^3=1,yxy^{-1}=x^2 \rangle,\ \langle x \rangle is a normal subgroup. Let G be a group with a subgroup H . Let G be generated by A\subseteq G . Suppose...
  24. R

    Is every Subgroup of a Cyclic Group itself Cyclic?

    Homework Statement Are all subgroups of a cyclic group cyclic themselves? Homework Equations G being cyclic means there exists an element g in G such that <g>=G, meaning we can obtain the whole group G by raising g to powers. The Attempt at a Solution Let's look at an arbitrary...
  25. A

    Proving that a subgroup is normal.

    Homework Statement How can we prove that a subgroup H of Gl_2(Z_3) is normal? These are the elements of H: \begin{pmatrix}1&1\\1&2 \end{pmatrix} \begin{pmatrix}1&2\\2&2 \end{pmatrix} \begin{pmatrix}2&1\\1&1 \end{pmatrix} \begin{pmatrix}2&2\\2&1 \end{pmatrix}...
  26. K

    Prove that Sym(F) is a subgroup of O2(R)

    Homework Statement \textbf{26.} Let F \subset \textbf{R$^2$} be a non-empty subset of \textbf{R$^2$} that is bounded. Prove that after chosing appropriate coordinates Sym(F) is a subgroup of O_2(\textbf{R}). Homework Equations The hints given are: Prove there is an a \in \textbf{R$^2$}...
  27. I

    MHB Use a subgroup lattice to compute a normalizer

    My question is at here: abstract algebra - Use a subgroup lattice to compute a normalizer - Mathematics Thank you!
  28. I

    MHB Identify isomorphism type for each proper subgroup of (Z/32Z)*

    The question is to identify isomorphism type for each proper subgroup of $(\mathbb{Z}/32\mathbb{Z})^{\times }$. (what's the "isomorphism type" means? Does the question mean we need to list all the ismorphism between of each subgroup and the respectively another group that is isomorphic to the...
  29. R

    Proving Finite Order Elements Form a Subgroup of an Abelian Group

    Homework Statement Prove the collection of all finite order elements in an abelian group, G, is a subgroup of G. The Attempt at a Solution Let H={x\inG : x is finite} with a,b \inH. Then a^{n}=e and b^{m}=e for some n,m. And b^{-1}\inH. (Can I just say this?) Hence...
  30. I

    MHB The union of an ascending chain of subgroups is a subgroup

    Let $G$ be a group, and $\left \{ H_{i} \right \}_{i\in \mathbb{Z}}$ be an ascending chain of subgroups of $G$; that is, $H_{i}\subseteq H_{j}$ for $i\leqslant j$. Prove that $\bigcup _{i\in \mathbb{Z}}H_{i}$ is a subgroup of $G$. I don't need the proof now. But can you show an example for me...
  31. G

    Proving a subgroup is equivalent to Z

    Homework Statement for n \in N, n \geq 1 Prove that (n^{3} +2n)Z + (n^{4}+3n^{2}+1)Z= Z Homework Equations I know subgroups of Z are of the form aZ for some a in Z and also that aZ+bZ= dZ, where d=gcd(a,b) The Attempt at a Solution So I was thinking if I could prove that the gcd...
  32. C

    Proving Subgroups in Finite Groups

    Homework Statement Let G be a finite group, a)Prove that if ##g\,\in\,G,## then ##\langle g \rangle## is a subgroup of ##G##. b)Prove that if ##|G| > 1## is not prime, then ##G## has a subgroup other than itself and the identity. The Attempt at a Solution a) This one I would just like...
  33. B

    How to prove Cs is a subgroup of C3v?

    Homework Statement prove that Cs is a subgroup of C3v group Homework Equations The Attempt at a Solution There are only two elements in Cs group, E and C_sigma. C_sigma is plane reflection operator which does not seem to exist in C3v group. This leads to my question here.
  34. caffeinemachine

    MHB O(G)=56. Sylow 2 subgroup has all its elements of order 2.

    Let $G$ be a group of order $56$ having at least $7$ elements of order $7$. 1) Prove that $G$ has only one Sylow $2$-subgroup $P$. 2) All elements of $P$ have order $2$. The first part is easy since it follows that the number of Sylow $7$-subgroups is $8$. I got stuck on part 2. From part 1 we...
  35. T

    How Is U(n) Embedded in O(2n) as a Subgroup?

    Hi everybody, I hope that I chose the right Forum for my question. As the title might suggest, I am interested in the embedding of the Lie algebra of U(n) into the Lie Algebra of O(2n). In connection with this it would be interesting to understand the resulting embedding of U(n) in O(2n). I...
  36. caffeinemachine

    MHB Finite group of order 4n+2 then elements of odd order form a subgroup.

    Let $G$ be a finite group of order $4n+2$ for some integer $n$. Let $g_1, g_2 \in G$ be such that $o(g_1)\equiv o(g_2) \equiv 1 \, (\mbox{mod} 2)$. Show that $o(g_1g_2)$ is also odd. I found a solution to this recently but I think that solution uses a very indirect approach. Not saying that that...
  37. T

    Intersection of subgroups is a subgroup

    Homework Statement Suppose H and K are subgroups of G. Prove H intersect K is a subgroup of G. Homework Equations Suppose G is a group and H is a nonempty subset of G. Then H is a subgroup of G iff a,b ∈ H implies ab^-1 ∈ H. The Attempt at a Solution Suppose a and b elements of H intersect...
  38. J

    Subgroup of Finitely Generated Abelian Group

    Homework Statement Prove that any subgroup of a finitely generated abelian group is finitely generated. Homework Equations The Attempt at a Solution I've attempted a proof by induction on the number of generators. The case n=1 corresponds to a cyclic group, and any subgroup of a...
  39. A

    Prove a subgroup of G/H X G/K is isomorphic to G/(H intersect K)

    Homework Statement Suppose H and K are normal subgroups of G. Prove that G/H x G/K has a subgroup isomorphic to G/(H\capK) Homework Equations The Attempt at a Solution I was trying to find a homomorphism from G to G/H x G/K where G/(H\capK) is the kernal. Maybe something like if g is in H it...
  40. M

    Does every element of order 2 in a finite group have a complement in the group?

    Let G be a finite group. Suppose that every element of order 2 of G has a complement in G, then G has no element of order 4. Proof. Let x be an element of G of order 4. By hypothesis, G=<x^{2}> K and < x^{2}> \capK=1 for some subgroup K of G. Clearly, G=< x> K and < x>\cap K=1$, but |G|=|<...
  41. G

    Finding permutations of a stabilizer subgroup of An

    Alright, I understand what a stabilizer is in a group, and I know how to find the permutations of An for any small integer n, but for a stabilizer, since it just maps every element to 1, would all permutations just be (1 2) (1 3) ... (1 n) for An?
  42. STEMucator

    Proving a Subgroup: Homework Statement

    Homework Statement I got this question from contemporary abstract algebra : http://gyazo.com/7a9e3f0603d1c0dcfde256e7b05276cd Homework Equations One step subgroup test : 1. Find my defining property. 2. Show that my potential subgroup is non-empty. 3. Assume that we have some a and b in our...
  43. srfriggen

    Determine the subgroup lattice for Z8

    Homework Statement "Determine the subgroup lattice for Z8" Homework Equations <1>={1,2,3,4,5,6,7,0} <2>={2,4,6,0} <3>=<1>=<5>=<7> <4>={4,0} <6>={6,4,2,0}The Attempt at a Solution My book only mentions this topic in one sentence and shows a diagram for Z30, which looks like a cube. I don't...
  44. srfriggen

    Abstract Algebra: List elements of Subgroup

    Homework Statement List the elements of the subgroups <3> and <7> in U(20). Homework Equations The Attempt at a Solution U(20)= {1, 3, 7, 9, 11, 13, 17, 19} = <3> = <7>. So basically I have that the common elements of, <3> and <7> and U(20), under + modulo 20, are all...
  45. J

    If H and K are subgroups of G, and K is normal, then HK is a subgroup of G.

    Ignoring the fact that it is redundant at times, is this proof correct? Also, is there a way to show that same result using the fact that K is closed with respect to conjugates rather than the fact that for all a in G, aK=Ka. Thank you! :) Proposition: If H and K are subgroups of G...
  46. J

    The GCD forms a subgroup of the integers

    Let r and s be positive integers. Show that {nr + ms | n,m ε Z} is a subgroup of Z Proof: ---- "SKETCH" ----- Let r , s be positive integers. Consider the set {nr + ms | n,m ε Z}. We wish to show that this set is a subgroup of Z. Closure Let a , b ε {nr + ms | n,m ε...
  47. C

    Is H a Free Commutative Group of Rank n in Z^n?

    show that H is subgroup of finite index in Z^n exactly when H is free comutative group of rank n
  48. E

    Is A3 a Normal Subgroup of S3?

    H=A3= {(1),(1 2 3),(1 3 2)} and G=S3 ={ (1),(1 2 3),(1 3 2),(1 2 ),(1 3),(1 2 3) } Is H is normal subgroup of G ? I try g=(1 2 3 ) for gH=Hg but gH≠Hg for all g ε G.In this situation,H is normal subgroup pf G?
  49. E

    All Subgroups of S3: Lagrange's Theorem Explained

    The question wants all subgroups of S3 . If H≤S 3 , then ; IHI=1,2,3,6 by Lagrance's Theorem. In other words, order of H can be 1,2,3 and 6. What ı want to ask is how to write subgroup of S3. For example,is H 1 (1) ?
  50. S

    Any group of order 952 contains a subgroup of order 68?

    Homework Statement I am struggling with a proof for this. Obviously Sylow's theorems come into play. We have that |G| = 952. As sylow's first theorem only covers subgroups of order pn, we cannot directly use it to assert the existence of a subgroup of order 68. On the other hand, if we can...
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