What is Subgroup: Definition and 289 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. P

    I Proof for subgroup -- How prove it is a subgroup of Z^m?

    Hi together! Say we have ## \Lambda_q{(A)} = \{\mathbf{x} \in \mathbb{Z}^m: \mathbf{x} = A^T\mathbf{s} \text{ mod }q \text{ for some } \mathbf{s} \in \mathbb{Z}^n_q\} ##. How can we proof that this is a subgroup of ##\mathbb{Z}^m## ? For a sufficient proof we need to check, closure...
  2. PhysicsRock

    Prove relation between the group of integers and a subgroup

    So, a friend of mine has attempted a solution. Unfortunately, he's having numbers spawn out of nowhere and a lot of stuff is going on there which I can't make sense of. I'm going to write down the entire attempt. $$ 0 \in X \; \text{otherwise no subgroup since neutral element isn't included}...
  3. penroseandpaper

    I Subgroup axioms for a symmetric group

    Hi, The textbook asks me to use subgroup axioms to prove why a set of permutations that interchange two specific symbols in S4 is or isn't a subgroup of the symmetric group, and the same for a set of permutations that fix two elements. My guess is that the set of permutations that interchange...
  4. R

    Show N is a normal subgroup and G/N has finite element

    Clearly e ∈ N. If a, b ∈ N, say ##a^k = b^l = e##, for some k,l ∈ N, then ##(ab)^{kl} = (a^k )^l (b^l )^k = e^l e^k = e##; thus, ab ∈ N. Also, ##|a|=|a^{−1}|##, so ##a^{−1}## ∈ N. Thus, N is a subgroup. As G is abelian, it is normal. Take any c ∈ G. If, for some n ∈ N, we have ##(cN)^n = eN##...
  5. L

    A Difference Between Subgroup & Closed Subgroup of a Group

    What is difference between subgroup and closed subgroup of the group? It is confusing to me because every group is closed. In a book Lie groups, Lie algebras and representations by Brian C. Hall is written "The condition that ##G## is closed subgroup, as opposed to merely a subgroup, should be...
  6. binbagsss

    Elliptic functions, properties of periods, discrete subgroup

    Homework Statement HiI am following this proof attached and am just stuck on the bit that says: ‘since ##\Omega## is a group it follows that ##|z-\omega|<2\epsilon ## contains..’Tbh, I have little knowledge on groups , it’s not a subject I have really studied in any of my classes-so the only...
  7. Prof. 27

    Finding Cosets of subgroup <(3,2,1)> of G = S3

    Homework Statement Find all cosets of the subgroup H in the group G given below. What is the index (G : H)? H = <(3,2,1)>, G = S3 Homework EquationsThe Attempt at a Solution I will leave out the initial (1,2,3) part of the permutation. We have S3 =...
  8. Mr Davis 97

    Showing that subgroup of unique order implies normality

    Homework Statement Let ##H## be a subgroup of ##G## and fix some element ##g\in G##. Prove that ##gHg^{-1}=\{ghg^{-1} \mid h\in H\}## is a subgroup of ##G## of the same order as ##H##. Deduce that if ##H## is the unique subgroup of ##G## of order ##|H|## then ##H\trianglelefteq G##. Homework...
  9. H

    MHB Question on subgroup and order of the elements

    Let G be the group of symmetries (including flips) of the regular heptagon (7-gon). As usual, we regard the elements of G as permutations of the set of vertex labels; thus, G ≤ S7. (a) Let σ denote the rotation of the 7-gon that takes the vertex 1 to the vertex 2. Write down the cyclic...
  10. Mr Davis 97

    Normal subgroup generated by a subset A

    Homework Statement Let ##G## be a group and let ##A \subseteq G## be a set. The normal subgroup of ##G## generated by ##A##, denoted ##\langle A \rangle ^N##, is the set of all products of conjugates of elements of ##A## and inverses of elements of ##A##. In symbols, $$\langle A \rangle ^N= \{...
  11. K

    I Proving that a subgroup is normal

    In this PDF, http://www.math.unl.edu/~bharbourne1/M417Spr04/M417Exam2Solns.pdf, in answering why a subgroup of index 2 is normal, the author says that the only two cosets must be ##A## and ##gA##. Why so? Why there can't be another element ##g'## such that ##G = g' A + g A##?
  12. N

    Proving Subgroup of Z/3Z

    Homework Statement I am looking at the quotient group G = Z/3Z which is additive and abelian. The equivalence classes are: [0] = {...,0,3,6,...} [1] = {...,1,4,7,...} [2] = {...,2,5,8,...} I want to prove [0] is a normal subgroup, N, by showing gng-1 = n' ∈ N for g ∈ G and n ∈ N. Since G...
  13. Mr Davis 97

    Showing that preimage of a subgroup is a subgroup

    Homework Statement Prove that if ##f:G\to H## is a group homomorphism and ##K\leq H## then the preimage of ##K##, defined as ##f^{-1}(K)=\{g\in G | f(g)\in K\}##, is a subgroup of ##G##. Homework EquationsThe Attempt at a Solution 1) Note that ##f^{-1}(K)## is nonempty, since ##f(e_G) = e_H##...
  14. Mr Davis 97

    Showing that a subgroup of Sym(4) is isomorphic to D_8

    Homework Statement Let ##R## be the set of all polynomials with integer coefficients in the independent variables ##x_1, x_2, x_3, x_4##. ##S_4## acts on ##R## by the group action ##\sigma \cdot p(x_1,x_2,x_3,x_4) = p(x_{\sigma(1)},x_{\sigma(2)},x_{\sigma(3)},x_{\sigma(4)})##. Exhibit all...
  15. Mr Davis 97

    I Using group action to prove a set is a subgroup

    Problem: Let ##G=S_n##, fix ##i \in \{1,2, \dots, n \}## and let ##G_i = \{ \sigma \in G ~|~ \sigma (i) = i \}##. Use group actions to prove that ##G_i## is a subgroup of G. Find ##|G_i|##. So here is what I did. Let ##A = \{1,2, \dots, n \}##. I claim that ##G## acts on ##A## by the group...
  16. Mr Davis 97

    If H is a subgroup, then H is subgroup of normalizer

    Homework Statement Show that if ##H## is a subgroup of ##G##, then ##H \le N_G (H)## Homework EquationsThe Attempt at a Solution Essentially, we need to show that ##H \subseteq N_G (H)##; since they are both groups under the same binary operation the fact that they are subgroups will result...
  17. Mr Davis 97

    Show that union of ascending chain of subgroups is subgroup

    Homework Statement Let ##H_1 \le H_2 \le \cdots## be an ascending chain of subgroups of ##G##. Prove that ##H = \bigcup\limits_{i=1}^{\infty} H_{i}## is a subgroup of ##G##. Homework EquationsThe Attempt at a Solution Certainly ##H## is nonempty, since each subgroup ##H_i## has at least the...
  18. Mr Davis 97

    Showing that upper triangular matrices form a subgroup

    Homework Statement Let ##n \in \mathbb{Z}^+## and let ##F## be a field. Prove that the set ##H = \{(A_{ij}) \in GL_n (F) ~ | ~ A_{ij} = 0 ~ \forall i > j \}## is a subgroup of ##GL_n (F)## Homework EquationsThe Attempt at a Solution So clearly the set is nonempty since ##I_n## is upper...
  19. Mr Davis 97

    Conditions on H and K if H ∪ K is a subgroup

    Homework Statement Let H and K be subgroups of G. Prove that if ##H \cup K## is a subgroup of ##G## then ##H \subseteq K## or ##K \subseteq H## Homework EquationsThe Attempt at a Solution Suppose that ##H \cup K \le G##. For contradiction, suppose that neither H nor K is a subset of the...
  20. B

    Subgroup of Index ##n## for every ##n \in \Bbb{N}##.

    Homework Statement A nonzero free abelian group has a subgroup of index ##n## for every positive integer ##n## Homework EquationsThe Attempt at a Solution If ##F## is a nonzero free abelian group, then ##F## is isomorphic to the direct sum ##G= \sum_{i \in I} \Bbb{Z}##, where ##I \neq...
  21. K

    I Issue regarding a subgroup

    I was reading a Wikipedia page where it's given an example of a group that's not a Lie Group. Here's the page https://en.wikipedia.org/wiki/Lie_group ; refer to "Counterexample". If we work with the topology of ##\mathbb{T}^2## it seems obvious that a map from some ##\mathbb{R}^m## would not be...
  22. F

    Subgroup of an arbitrary group

    Homework Statement Let G be a group. Let H and K be subgroups of G. Prove that if H ##\subseteq## K, then H is a subgroup of K. Homework EquationsThe Attempt at a Solution H is a subset of K and H,K are groups. if x,y, xy ##\epsilon## H, then x,y, xy ##\epsilon## K. So H is closed under...
  23. J

    MHB Proving K is a Subgroup of G: Subgroup Nesting in H and L

    Let H be a subgroup of G and let L be a subgroup of H. Prove that K is a subgroup of G. This question seems very redundant to me, isn't anything in a subgroup automatically a subgroup of anything the larger group is a subgroup of. Can some one explain this proof to me?
  24. J

    MHB Proving N(H) is a Subgroup of G Containing H

    Let G be a group and let H be a subgroup. Define N(H)={x∈G|xhx-1 ∈H for all h∈H}. Show that N(H) is a subgroup of G which contains H. To be a subgroup I know N(H) must close over the operations and the inverse, but I am not sure hot to show that in this case.
  25. Mr Davis 97

    Showing that dihedral 4 is isomorphic to subgroup of permutations

    Homework Statement D4 acts on the vertices of the square. Labeling them counterclockwise starting from the top left as 1, 2, 3, 4, find the corresponding homomorphism to S4. Homework EquationsThe Attempt at a Solution I am not completely sure what the question is asking. It's pretty clear to...
  26. Mr Davis 97

    I Generalizing the definition of a subgroup

    Let ##G## be a group. I have shown that ##H_a = \{x \in G | xa=ax \}## is a subgroup of G, where ##a## is one fixed element of ##G##. I am now asked to show that ##H_S = \{x \in G ~| ~xs=sx,~ \forall s \in S\}## is a subgroup of ##G##. How would proving the former differ from proving the latter...
  27. binbagsss

    I Definition of discrete Subgroup quick q

    Hello, Just a really quick question on definition of discrete subgroup. This is for an elliptic functions course, I have not done any courses on topology nor is it needed, and most of the stuff I can see online refer to topology alot, so I thought I'd ask here. I need it in the complex plane...
  28. Kara386

    Group theory -- show H is a subgroup of O(2)

    Homework Statement Let ##R(\theta) = \left( \begin{array}{cc} \cos(\theta) & -\sin(\theta)\\ \sin(\theta)& \cos(\theta)\\ \end{array} \right) \in O(2)## represent a rotation through angle ##\theta##, and ##X(\theta) = \left( \begin{array}{cc} \cos(\theta) & \sin(\theta)\\ \sin(\theta)&...
  29. G

    Prove (Q+, *) is isomorphic to a proper subgroup of itself

    Homework Statement Prove that Q+, the group of positive rational numbers under multiplication, is isomorphic to a proper subgroup of itself. Homework Equations None The Attempt at a Solution [/B] Not at all sure if this is legit. Let phi: Q+ --> G phi(x) = x2, x is in Q+ We will...
  30. M

    MHB Subgroup of the Galois group

    Hey! :o Let $\rho=\sqrt[3]{\frac{1+\sqrt{5}}{2}}$. We have that $\rho$ is a root of $f(x)=x^6-x^3-1\in \mathbb{Q}[x]$, that is irreducible over $\mathbb{Q}$. We have that all the roots of $f(x)$ are $\rho, \omega\rho, \omega^2\rho, -\frac{1}{\rho}, -\frac{\omega}{\rho}...
  31. M

    MHB The extension is Galois iff H_i is a normal subgroup of H_{i-1}

    Hey! :o Let $E/F$ be a finite Galois extension and let the chain of extensions $F = K_0 \leq K_1 \leq \dots \leq K_n = E$. Let $G = Gal(E/F)$ and, for $i = 0, 1, \dots , n$, let $H_i$ be the subgroup of $G$, that corresponds to $K_i$ through the Galois mapping. I want to show that, for any...
  32. S

    I Normalizer of a subgroup of prime index

    Hello! Can anyone help me with this problem? If H is a subgroup of prime index in a finite group G, show that either N(H)=G or N(H) = H. Thank you!
  33. S

    I Notation N(H) for a subgroup

    Hello! I have this problem: If H is a subgroup of prime index in a finite group G, show that either H is a normal subgroup or N(H) = H. What does N(H) means? I don't want a solution for the problem (at least not yet), I just want to know what that notation means. Thank you!
  34. RJLiberator

    Proving that Aut(G) is a subgroup of Bij(G)

    Homework Statement Let G be a group. An Isomorphism Φ: G --> G is called an automorphism of G. Let Aut(G) denote the set of all automorphisms of G. Prove that Aut(G) is a subgroup of Bij(G). Homework Equations For it to be a subgroup we need to show: i) e ∈ Aut(G) ii) For all x,y ∈ Aut(G)...
  35. M

    MHB There is no proper subgroup

    Hey! :o Let $P$ be a $p$-Sylow subgroup in $G$ and $N=N_G(P)$. I want to show that there is no proper normal subgroup $H$ of $G$ that contains $N$. We suppose that there is a proper normal subgroup $H$ of $G$ that contains $N$, $$N\leq H<G$$ Then $[G:N]=[G:H][H:N]$, with $[G:H]>1$. How...
  36. M

    MHB G contains a normal p-Sylow subgroup

    Hey! :o Let $G$ be a non-abelian finite group with center $Z>1$. I want to show that if $G/Z$ is a $p$-group, for some prime $p$, then $G$ contains a normal $p$-Sylow subgroup and $p\mid |Z|$. We have that $$|G/Z|=p^n, n\geq 1\Rightarrow \frac{|G|}{|Z|}=p^n\Rightarrow |G|=p^n|Z|$$ That means...
  37. M

    I Sylow subgroup of some factor group

    Hi. I have the following question: Let G be a finite group. Let K be a subgroup of G and let N be a normal subgroup of G. Let P be a Sylow p-subgroup of K. Is PN/N is a Sylow p-subgroup of KN/N? Here is what I think. Since PN/N \cong P/(P \cap N), then PN/N is a p-subgroup of KN/N. Now...
  38. M

    MHB Show that the intersection is a pp -Sylow subgroup

    Hey! :o I want to show that if $S\in \text{Syl}_p(G)$ and $N\trianglelefteq G$, then $N\cap S\in \text{Syl}_p(N)$. Could you give me some hints how we could show that? (Wondering) Do we maybe use Frattini's Argument? (Wondering) From that we have that since $N\trianglelefteq G$ and $S\in...
  39. NoName3

    MHB Find Left Cosets of Subgroup in $\mathbb{Z}_{15}, D_4$

    How do I find the left cosets of: $(a)$ $\left\{ [0], [5], [10] \right\} \le \mathbb{Z}_{15}$ ($\mathbb{Z}_n$ is additive group modulo $n$). $(b)$ $\left\{e, y, y^2, y^3 \right\} \le D_4$ where $y$ denotes rotation of a square. The not equal to here denotes subgroup. The trouble I've with...
  40. S

    I Why only normal subgroup is used to obtain group quotient

    Hello! As far as I know any subgroup can, in principle, be used to divide group into bundle of cosets. Then any group element belongs to one of the cosets (or to the subgroup itself). And such division still is not qualified as a quotient. Yes, the bundle of cosets in this case will be...
  41. G

    MHB Verifying Subgroup Notation of $H(x_0)$ in $A(S)

    Let $S$ be any set, $A(S)$ the set of one-to-one mappings of $S$ onto itself, made into a group under the composition of mappings. If $x_0 \in S$, what is meant by $H(x_0) = \left\{\phi \in A(S): x_0 \phi = x_0\right\}$? The set that contains the element $\phi$ in $A(S)$ that maps $x_0$ onto...
  42. M

    MHB Show that it is a normal subgroup of S4

    Hey! :o I want to show that $N\{1, (12)(34), (13)(24), (14)(23)\}$ is a normal subgroup of $S_4$ that is contained in $A_4$ and that satisfies $S_4/N\cong S_3$ and $A_4/N\cong Z_3$. Let $\sigma\in S_4$. We have the following: $$\sigma 1 \sigma^{-1}=\sigma (1) \\ \sigma (1 2)(3 4)...
  43. M

    MHB Proof: Every Subgroup of Cyclic $H$ is Normal in $G$

    Hey! :o I want to show that if $H$ is a cylic normal subgroup of a group $G$, then each subgroup of $H$ is a normal subgroup of $G$. I have done the following: Since $H$ is a normal subgroup of $G$, we have that $$ghg^{-1}=h\in H, \ \forall g \in G \text{ and } \forall h\in H \tag 1$$...
  44. mnb96

    Elements of semigroup commuting with subgroup

    Hello, Suppose we have a semigroup S with a subgroup G≤S. Assume there is an element s∈S that commutes with all the elements in G. Does this statement implies (or is equivalent to) another statement? If hypothetically the element s would have been in G, then we could have said that s was an...
  45. I

    Prove that T(G) is subgroup of G

    Homework Statement Homework Equations subgroup axioms: 1. a, b in T(G), then ab in T(G) 2. existence of identity element. 3. a in T(G), then a^-1 in T(G) The Attempt at a Solution 1. let a be in T(G), then a^n = e. let b be in T(G), then b^n = e (ab)^n = (a^n)(b^n) = (e)(e) = e axiom 1...
  46. J

    Show group equivalence relation associated with normal subgroup

    Homework Statement Let ##G## be a group and ##\sim## and equivalence relation on ##G##. Prove that if ##\sim## respects multiplication, then ##\sim## is the equivalence relation associated to some normal subgroup ##N\trianglelefteq G##; i.e., prove there is a normal subgroup ##N## such that...
  47. Q

    Show that a normal subgroup <S> is equal to <T>

    Note: I only need help on the underlined portion of the problem, but I'm including all parts since they may provide relevant information. Thanks in advance. 1. Homework Statement Let S be a subset of a group G such that g−1Sg ⊂ S for any g∈G. Show that the subgroup ⟨S⟩ generated by S is...
  48. R

    Finding a normal subgroup H of Zmn of order m

    Homework Statement Find a normal subgroup H of Zmn of order m where m and n are positive integers. Show that H is isomorphic to Zm. Homework EquationsThe Attempt at a Solution I am honestly not even sure where to start. My initial thoughts were if Zmn was isomorphic to Zm x Zn then I could...
  49. PsychonautQQ

    Why is the core of a subgroup contained in the subgroup?

    Let H be a subgroup of G, then: Core H = {a in G | a is an element of gHg^(-1) for all g in G} = The intersection of all conjugates of H in G My book goes on to say that every element of Core H is in H itself because H is a conjugate to itself. Previously, I understood that H was a conjugate to...
  50. 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...
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