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. R

    Order of subgroup G - representing triangular prism

    Please see attached diagram here is what I have done in order to answer this question Triangular prism above represents the group G of all symmetries of the prism as permutation of the set {123456} Part a: is to describe geometrically the symmetries of the prism represented in the cycle...
  2. F

    How to find subgroup of index n in a given group

    Dear Folks: Is there a general method to find all subgroups in a given abstract group?? Many Thanks! This question came into my classmates' mind when he wants to find a 2 sheet covering of the Klein Bottle. This question is equivalent to find a subgroup of index 2 in Z free product...
  3. D

    Exploring Semipermutable Subgroups in S4 for Scientists

    How i can find the semipermutable subgroups in S4? i konw that the normal subgroup is semipermutable .
  4. L

    What Are the Rules for Element and Subgroup Orders in Group Theory?

    Hey, I'm just trying to grasp ordering of groups and subgroups a little better, I get the basics of finding the order of elements knowing the group but I have a few small questions, If you have a group of say, order 100, what would the possible orders of an element say g^12 in the...
  5. L

    Homomorphism of a cyclic subgroup is a cyclic subgroup ?

    Homework Statement Let \alpha:G \rightarrow H be a homomorphism and let x\inG Prove \alpha(<x>) =<\alpha(x)> Homework Equations α(<x>) = α({x^{r}: r ∈ Z}) = {α(x^{r}) : r ∈ Z} = {α(x)^{r}: r ∈ Z} = <α(x)>. I do not understand how can we take out the 'r' out of a(x^{r}) to...
  6. J

    Is H a Subgroup of G in an Abelian Group?

    Homework Statement Let <G, *> be an Abelian group with the identity element, e. Let H = {g ε G| g2 = e}. That is, H is the set of all members of G whose squares are the identity. (i) Prove that H is a subgroup of G. (ii) Was being Abelian a necessary condition? Homework Equations For...
  7. T

    Subgroup conjugation and cosets

    Hello, I am having trouble with the following problem. Suppose that H is a subgroup of G such that whenever Ha≠Hb then aH≠bH. Prove that gHg^(-1) is a subset of H. I have tried to manipulate the following equation for some ideas H = Hgg^(-1) = gg^(-1)H but I don't know how to go...
  8. GreenGoblin

    MHB Subgroups of a Cyclic Group G of Order 12

    Determine the subgroups Let S: = {(123),(235)} be a subset of $\sum_{5}$. Determine the subgroup <S> of $\sum_{5}$? What exactly is this asking? Is S the set of just two elements here? Or all elements containing these permutations? If the former how can S be a subgroup? Let G = <g> be a...
  9. O

    Normal subgroup with prime index

    Homework Statement Prove that if p is a prime and G is a group of order p^a for some a in Z+, then every subgroup of index p is normal in G. Homework Equations We know the order of H is p^(a-1). H is a maximal subgroup, if that matters. The Attempt at a Solution Suppose H≤G and...
  10. P

    G\K: Is G\K a Subgroup? | Index 2 Group and Subgroup K

    Lets say I have a group of index 2 and I have some subgroup K. Is it that G\K is a subgroup as well?
  11. C

    Abstract Algebra - Smallest subgroup of GL(n,R)

    Homework Statement Let A = \left[ \begin{array}{cccc} 1 & 1 \\ 0 & -1 \end{array} \right] Let B = \left[ \begin{array}{cccc} 1 & 2 \\ 0 & -1 \end{array} \right] Find the smallest subgroup G of GL(n,R) that contains A and B. Also, find the smallest subgroup H of G that contains the matrices...
  12. L

    Normal Subgroup Conjugate of H by element

    Homework Statement Let H be a subgroup of group G. Then H \unlhd G \Leftrightarrow xHx^{-1}=H \forall x\in G \Leftrightarrow xH=Hx \forall x\in G \Leftrightarrow xHx^{-1}=Hxx^{-1} \forall x\in G \Leftrightarrow xHx^{-1}=HxHx^{-1}=H \forall x\in G...
  13. F

    What is the purpose of <-i> in a complex number subgroup?

    Well I generally haven an idea about subgroup of a group and generators. But I fail to understand following: ({1,-1,i,-i},X) I can see <1>={1} <-1>={-1,1} <i>={i,-1,-i,1} But simply have no idea about <-i> How can you work with -i?
  14. B

    Why is the smallest subgroup of G containing A and B equal to G itself?

    In an example it says that, if |G| = 15 and G has subgroups A,B of G with |A| = 5 and |B| = 3 , then A \cap B must equal \{e_G\} and the smallest subgroup of G containing both A and B is G itself. Could anyone explain why? Thanks!
  15. P

    Prove H is a Subgroup of G: ab=ba

    If ab=ba in a group G, let H={g\inG|agb=bga}. Show that H is a subgroup.I have the identity because ab=ba implies that aeb=bea\inH. I can't seem to figure out how to get the inverses or closer. Any suggestions or slight nudges in the right direction?
  16. P

    Subgroup of A4: Even Permutations on 4 Elements

    let M={x^{2}|x\inG} where G is a group. Show M is not a sub group if G=A_{4}(even permutations on 4 elements.) (Since an even times an even is even its closed, the identity is even and all the inverses should be even.)* Is is associativity were it is going to go wrong? Or am I wrong in thinking *.
  17. F

    Is N3 a Subgroup of Dihedral Group Dih(12)?

    Homework Statement Taking the Dih(12) = {α,β :α6 = 1, β2 = 1, βα = α-1β} and a function Nr = {gr: g element of Dih(12)} Homework Equations Taking the above I have to find the elements of N3. And then prove that N3 is not a subgroup of Dih(12). The Attempt at a Solution For N3 I...
  18. M

    Maximal subgroup of a product of groups?

    Let G and H be finite groups. The maximal subgroups of GxH are of the form GxM where M is maximal subgroup of H or NxG where N is a maximal subgroup of G. Is this true?
  19. B

    Is a Subgroup of a Normal Subgroup Always Normal?

    If N is normal in G, is its normal quotient (factor) group G/N normal as well? And does it imply that any subgroup of G/N which has a form H/N is normal as well? Is a subgroup of a normal subgrp normal as well. It appears that not always. Thanks
  20. M

    Is <(12)> a Maximal Subgroup of S_{3}?

    If G is a finite group and M is a maximal subgroup, H is a subgroup of G not contained in M. Then G=HM. Is this true?
  21. A

    Prove that N is a normal subgroup

    Homework Statement If H is any subgroup of G and N={\cap_{a\in G} a^{-1}Ha}, prove that N is a normal subgroup of G. The Attempt at a Solution Is this statement true? \forall n: n \in N \implies \exists h \in H : n=a^{-1}ha The theorem looks intuitively true, but I don't know how to...
  22. M

    What is the Normalizer of a Subgroup?

    Homework Statement Prove the number of distinct conjugate subgroups of a subgroup H in a group G is [G:N(H)] where N(H)={g \in G | gHg^{-1}=H}. Homework Equations I'm thinking the counting formula; G=|C(x)||Z(x)| with C(x) being the conjugacy class of x and Z(x) being the centralizer...
  23. D

    If a group G has order p^n, show any subgroup of order p^n-1 is normal

    If a group G has order p^n , show that any subgroup of order p^{n-1} is normal in G. i have no idea how to start this. i know that to show that a subgroup N is normal in G, i need that gNg^{-1} = N . so i start with any subgroup N of order p^{n-1} but i have no idea how to continue. this...
  24. H

    Proving H is a subgroup of G, given

    Homework Statement Let G be an abelian group, k a fixed positive integer, and H = {a is an element of G; |a| divides k}. Prove that H is a subgroup of G. Homework Equations Definition of groups, subgroups, and general knowledge of division algorithm. The Attempt at a Solution I...
  25. S

    Left coset of a subgroup of Complex numbers.

    Homework Statement For H \leq G as specified, determine the left cosets of H in G. (ii) G = \mathbb{C}* H = \mathbb{R}* (iii) G = \mathbb{C}* H = \mathbb{R}_{+}The Attempt at a Solution I have the answers, it's just a little inconsistency I don't understand. For (ii) left cosets are...
  26. L

    Kernal of a Subgroup: Explained

    I'm having a bit of a problem understanding the kernal of a subgroup. It appears to always be equal to the identity element. But that doesn't seem to make much sense. Anyone care to clarify this for me?
  27. ArcanaNoir

    Isomorphic groups G and H, G has subgroup order n implies H has subgroup order n

    Homework Statement G is isomorphic to H. Prove that if G has a subgroup of order n, H has a subgroup of order n. Homework Equations G is isomorphic to H means there is an operation preserving bijection from G to H. The Attempt at a Solution I don't know if this is the right...
  28. I

    Proof of Cyclic Subgroup Equivalence for Finite Groups

    Homework Statement Suppose a \in <b> Then <a> = <b> iff a and b have the same order (let the order be n - the group is assumed to be finite for the problem). Proof: Suppose a and b have the same order (going this direction I'm trying to show that <a> is contained in <b> and <b> is...
  29. ArcanaNoir

    Index of subgroup H is 2 implies

    Index of subgroup H is 2 implies... Homework Statement Just had my abstract algebra test. This was the only question I did not answer. The rest I answered somewhat confidently. Prove that if H is a subgroup of G, [G : H] = 2, a,b are in G but not in H, then ab is in H. Homework...
  30. ArcanaNoir

    Proof involving group and subgroup

    Homework Statement If H is a subgroup of a group G, and a, b \in G, then the following four conditions are equivalent: i) ab^{-1} \in H ii) a=hb for some h \in H iii) a \in Hb iv) Ha=Hb Homework Equations cancellation law seems handy, and existence of an inverse, associativity, and...
  31. H

    Order of subgroup of an abelian group

    Homework Statement Suppose H and K are subgroups of an abelian group G (not neccessarily finite). Let the order of H and K be a and b respectively. Prove that there exists a subgroup of order L, where L = lcm(a,b). Homework Equations Product Formula: |HK|/|H| = |K|/|H intersect K|...
  32. S

    Abstract Algebra: Subgroup Proof

    Homework Statement Show that if H is a subgroup of G and K is a subgroup of H, then K is a subgroup of G. Homework Equations The Attempt at a Solution Well I know that H is a subgroup of G if H is non empty, has multiplication, and his inverses. So I assume that K is a subgroup...
  33. D

    Showing that a certain subgroup is the kernal of a homomorphism

    Let Q be the quaternion group {1, -1, i, -i, j, -j, k, -k}. Show that the normal subgroup {1, -1, i, -i} is the kernal of a homomorphism from Q to {1, -1}. I know that if N is a normal subgroup of G then the homomorphism f: G -> G/N has N as the kernal of f. while i can get the kernal of f to...
  34. I

    Abstract Algebra - Subgroup of Permutations

    Homework Statement A is a subset of R and G is a set of permutations of A. Show that G is a subgroup of S_A (the group of all permutations of A). Write the table of G. Onto the actual problem: A is the set of all nonzero real numbers. G={e,f,g,h} where e is the identity element...
  35. E

    Advanced Algebra: Non-Finitely GEnerated Subgroup

    Hello everyone, I'm really lost on where to start this problem; any thoughts? Thanks...
  36. E

    A particular subgroup of a Free Group is normal

    Hello friends, I'm working through my book and I'm having a lot of trouble coming to terms / believing this. Could anyone assist? Let F be a free group and N be the subgroup generated by the set {x^n : x is in F and n is fixed} then N is normal in F. Any ideas?
  37. I

    Proving H as a Subgroup of G: Using the Abelian Property

    Homework Statement G is an abelian group Let H = {x \in G : x = x^{-1} Prove H is a subgroup of G. I have two methods in my arsenal to do this (and I am writing them out additively just for ease): 1. Let a,b be in H. If a + b is in H AND -a is in H then H<G. or 2.Let a,b be in H...
  38. S

    Is H a Subgroup of N if |H| and |G:N| are Relatively Prime?

    Let G be a finite group, let H be a subgroup of G and let N be a normal subgroup of G. Show that if |H| and |G:N| are relatively prime then H is a subgroup of N. I have tried using the fact that since N is normal, HN is a subgroup of G. Suposing that H is not contained in N, I tried finding...
  39. S

    Proof that a certain group G contains a cyclic subgroup of order rs

    Homework Statement Let G be an abelian group and let H and K be finite cyclic subgroups with |H|=r and |K|=s. Show that if r and s are relatively prime, then G contains a cyclic subgroup of order rsHomework Equations Fundemental theorem of cyclic group which states that the order of any...
  40. N

    Same symbol, different meanings? Subgroup index & field extension

    Hello, Say we have field (F,+,.) and field extension (E,+,.), then the degree of the field extension (i.e. the dimension of the vector field E across the field F) is given the symbol [E:F] . But we can also see F and E as the groups (F,+) and (E,+), and then the same symbol denotes the...
  41. O

    Every subgroup of index 2 is normal?

    I have to prove that every subgroup H of a group G with index(number of distinct cosets of the subgroup) 2 is normal. I don't know how to start :'( Please help.
  42. O

    Derived Subgroup (In particular Q8)

    So if G = Q8 = <a, b : a^4 = 1, b^2 = a^2, b^{-1}ab = a^{-1}> I'm fine with the notion of the derived subgroup G' = <[g,h] : g, h in G> (Where [g,h] = g^{-1}h^{-1}gh) But I can't see why G' = {1, a^2}, I can only seem to get everything to be 1!? i.e. g = a, h = a^3 ===> a^{-1}a^{-3}aa^3...
  43. S

    Checking if H is a subgroup of G

    Homework Statement Let G = GL_n(\mathbb{C}) and H = GL_n(\mathbb{R}) Homework Equations Prop. A subset H of a group G is a subgroup if: 1. If a,b \in H, then ab \in H. 2. 1 \in H 3. \forall a \in H \exists a^{-1} \in H. The Attempt at a Solution My intuition says the answer is yes since the...
  44. B

    Determining if Subset is a Subgroup by using Group Presentation

    Hi, Algebraists: Say I'm given a group's presentation G=<X|R>, with X a finite set of generators, R the set of relations. A couple of questions, please: i)If S is a subset of G what condition must the generators of S satisfy for S to be a subgroup of G ? I know there is a condition...
  45. A

    Subgroup Order in Groups of Divisible Orders: Proof or Counterexamples?

    If G is a group of order n, and n is divisible by k. Then must G have a subgroup of order k? proof or counterexamples?
  46. V

    Prove that a group of order 42 has a nontrivial normal subgroup

    Homework Statement Prove that a group of order 42 has a nontrivial normal subgroup Homework Equations We are supposed to use Cauchys Theorem to solve the problem We are not allowed to use any of Sylows Theorems The Attempt at a Solution By using Cauchys Theorem i know there...
  47. E

    Discrete quotient group from closed subgroup

    Hi All, I've come across a theorem that I'm trying to prove, which states that: The quotient group G/H is a discrete group iff the normal subgroup H is open. In fact I'm only really interested in the direction H open implies G/H discrete.. To a lesser extent I'm also interested in the H...
  48. Z

    Does G/H Form a Subgroup of G?

    Suppose G is a group and H is a subgroup of G. Then is G\H a subgroup itself? My feeling is it shouldn't be since 1\inH, therefore 1\notinG\H? I'm getting a bit confused about this because I'm doing a homework sheet and the question deals with a homomorphism from G \rightarrow G\H and I'm...
  49. I

    Setwise stabilizer of a finite set is a maximal subgroup of Sym(N)

    So I'm reading a paper which assumes the following statement but I would like to be able to prove it. Let S denote the symmetric group on the natural numbers. If \emptyset\subset A \subset \mathbb{N} then S_{\{A\}}=\{f\in S:af\in a,\;\forall{a}\in A\}$ is a maximal subgroup of S...
  50. M

    Let G be a group and H a subgroup. Prove if [G:H]=2, then H is normal.

    Let G be a group and H be a subgroup of G. Prove that if [G:H]=2, then H is a normal subgroup of G.
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