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. N

    Question on Automorphism group as subgroup

    The problem is as follows:- Statement of the problem: Given group G, show that the automorphism group of G is a subgroup of the permutation group of G. I can show that Auto(G) is a subset of Perm(G) easily. So I have to show that subgroup conditions hold: (1) for each x in Auto(G), x inverse...
  2. L

    Subgroup of a symmetric group Sn

    Homework Statement Show that if G is a subgroup of a symmetric group Sn, then either every element of G is an even permutation or else exactly half the elements of G are even permutations. Homework Equations The Attempt at a Solution We have a hint for the problem. If all the...
  3. X

    H is a subgroup of its Centralizer iff H is Abelian

    Homework Statement G is a group, and H is a subgroup of G. (1) Show H is a subgroup of its Normalizer. Give an example to show that this is NOT necessarily true if H is NOT a subgroup. (2) Show H is a subgroup of its Centralizer iff H is Abelian Homework Equations normalizer NG(H) = {g in...
  4. X

    Find the order of the cyclic subgroup of D2n generated by r

    Homework Statement Find the order of the cyclic subgroup of D2n generated by r. Homework Equations The order of an element r is the smallest positive integer n such that r^n = 1. Here is the representation of Dihedral group D2n = <r, s|r^n=s^2=1, rs=s^-1> The elements that are in D2n...
  5. M

    Need help in proving a subgroup result about the dyadic rationals

    Homework Statement Prove that every proper subgroup H of the dyadic rationals G (numbers of the form a/2^l, for integers a, l) which contains the integers is cyclic. Homework Equations The Attempt at a Solution I was trying to argue based on the 'proper' requirement, i.e. there is...
  6. R

    If ab is in a subgroup, are a and b neccessarily in the subgroup?

    Homework Statement I was just wondering if the product ab is in a subgroup, are a and b necessarily in the subgroup, as well? The Attempt at a Solution I think they are, but how would you prove that? Or is that obvious from closure under multiplication and you don't need to prove it? I know it...
  7. A

    Finite simple group with prime index subgroup

    Homework Statement If G is a finite simple group and H is a subgroup of prime index p Then 1. p is the largest prime divisor of \left|G\right| (the order of G) 2. p2 doesn't divide \left|G\right| I think I have this proved, but want to confirm my reasoning is sound. this problem is...
  8. R

    Z_2007 Subgroup of Order 6: True or False?

    Homework Statement True or false: U(\mathbb{Z}_{2007}), . (=group of units) has a subgroup of order 6. Homework Equations We know that the \phi(2007) (= Euler's tolient function) = 1332, which is the amount of elements in U(\mathbb{Z}_{2007}), . The Attempt at a Solution We clearly see...
  9. E

    Permutations of S_36; Subgroup

    Homework Statement Given Information: If sigma is a permutation of a set A, we say sigma moves "a" in set A iff sigma("a") is not equal to "a". For the symmetric group S_36 of all permutations of 36 elements, let H be a subset of S_36 containing all permutations that move no more than for...
  10. N

    Is H a Normal Subgroup If Every Left Coset Equals a Right Coset?

    Let G be a group and H be a subgroup of G. If every left coset xH, where x in G, is equal to a right coset Hy, for some y in G, prove H is normal subgroup. Help?
  11. J

    Anyone recognize this subgroup?

    Ahoy hoy, let A be a set with a \in A . Define G_a = \{ g \in S_A; g(a) = a \} Where S_A is the permutation group. Are we just talking the set of all inverses of the permutation group? Thanks!
  12. E

    Non-Abelian Subgroup Produced by

    This is not a homework question (although the answer will help answer a homework question). I know that a non-Abelian group can have both Abelian and non-Abelian subgroups but can a non-Abelian subgroup be produced by an Abelian group (or must the group be non-Abelian). Any thoughts...
  13. H

    Understanding the Automatic Formation of Lie Subgroups in a Lie Group

    Recently, I read the follow paragraph: Let $G$ be a Lie group. If $H$ is a subgroup defined by the vanishing of a number of (continuous) real-valued functions $$H=\{g\in G| F_i(g)=0, i=1,2,\cdots,n\},$$ then $H$ is automatically a Lie subgroup of $G$. We do not need to check the maximal...
  14. M

    Proving H is a Subgroup of R^*

    Homework Statement Let R^* be a group of all nonzero real numbers under multiplication and let H = {x in R^*: sqrt (x) is rational. Prove or disprove H is a subgroup of R^*. Homework Equations all axioms must be satisfied The Attempt at a Solution associative: satisfied. Both R^* and H have...
  15. C

    Is S7 x {0} a Maximal Normal Subgroup of S7 x Z7?

    1. Is the group S7 X {0} a maximal normal subgroup of the product group S7 X Z7 ? 2. No relevant equations 3. That kinda is my answer, original question was asking about S7 X Z7
  16. K

    Showing something is a subgroup of the direct product

    Homework Statement Let G1 and G2 be groups, with subgroups H1 and H2 respectively. Show that {(x1,x2) such that x1 is in H1, x2 is in H2} is a subgroup of the direct product G1 x G2 Homework Equations The Attempt at a Solution let G1, G2 be groups with H1, H2 subgroups. Let...
  17. J

    Is g^-1Ng a Subgroup of G? Proving Invariance in Group Theory

    Salutations all, just stuck with the starting step, I want to see if I can take it from there. Homework Statement Let G be a group and let N be a subgroup of G. Prove that the set g^{-1}Ng is a subgroup of G. The Attempt at a Solution Well, I'm going to have to show that...
  18. E

    Can S5 contain a subgroup of order 7?

    not homework question and i say yes.
  19. K

    Is 2a=0 a Subgroup of an Abelian Group?

    Homework Statement Let G be an abelian group such that the operation on G is denoted additively. Show that 2a=0 is a subgroup on G. Compute this subgroup for G=Z12. Homework Equations The Attempt at a Solution Well, I started out by knowing that abelian means ab=ba.
  20. K

    Showing something is a subgroup

    Homework Statement Let S be a set and let a be a fixed element of S. Show that s is an element of Sym(S) such that s(a)=a is a subgroup of Sym(S). Homework Equations The Attempt at a Solution
  21. K

    How to show a matrix is a subgroup of a group G.

    Homework Statement Let G=GL2(R) Show that T=matrix with row 1= a, b and row 2 = 0, d with ad\neq0 is a subgroup of G. Homework Equations The Attempt at a Solution I'm sort of confused on how to show it is a subgroup.
  22. J

    Cyclic subgroup of a particular form proof with the group infinitely large.

    Homework Statement G is additive group. If the order of G is infinity, then G is cyclic iff each subgroup H of G is of the form nG for some interger n. Homework Equations cyclic property. The Attempt at a Solution i kind of know that nG is the answer but why G has to be infinite...
  23. H

    Group of order pq && nonnormal subgroup

    I saw the following problem on my abstract algebra book (dummit && foote) , I tried to solve it but I couldn't : Let p, q be primes with p < q . Prove that a nonabelian group G of order pq has a nonnormal subgroup of index q , so there exists an injective homomorphism into Sq. Deduce that G...
  24. F

    "Does Finite Group Contain Subgroup of Index 2 if Element has Order 2?

    Is it true that if a finite group G contains a subgroup of index 2, then there is an element of G with order 2?
  25. A

    Meaning of colon in group theory, if not subgroup index?

    I am reading a paper where the author uses colons in the description of groups. Example (not verbatim): "This subgroup is isomorphic to (Z_5 X A_4):Z_2". Several subgroups are described in the same way (as (G_1 x G_2):G_3) throughout the paper. I have seen the colon in G:H to indicate the...
  26. K

    Is Using \leq for Subgroup Notation Incorrect?

    my lecturer use \leq for subgroup. For example H \leq S means H is a subgroup of S. But is it a wrong use of notation as the less-than-equal sign is about number?
  27. W

    Group automorphism not a subgroup?

    I was wondering if anyone could shed some light on this... I thought Aut(G) was always a subgroup of G but I don't think I can prove it. This is leading me to second guess this intuition. Could I get some reading reccomendations from anyone on this? Thx
  28. 1

    Proving an Abelian Group with 2 Elements of Order 2 has a Subgroup of Order 4

    Homework Statement Prove that an abelian group with two elements of order 2 must have a subgroup of order 4 Homework Equations The Attempt at a Solution Let G be an abelian group ==> for every a,b that belong to G ab=ba. Let a,b have order 2 ==> a^2 =e and b^2 = e. Since a...
  29. T

    Cyclic Subgroup H=<9> of Z30: List and Find Elements

    1. (a) List all elements in H=<9>, viewed as a cyclic subgroup of Z30 (b) Find all z in H such that H=<z> I'm thinking that H=<9> = {1,7,9} (viewed as a cyclic subgroup of Z30) is this correct? And could someone explain what (b) is asking in other terms?
  30. M

    Proving the Existence of One Cyclic Subgroup in a Cyclic Group of Order n

    Homework Statement Let G be a cyclic group of order n, and let r be an integer dividing n. Prove that G contains exactly one subgroup of order r. Homework Equations cyclic group, subgroup The Attempt at a Solution Say the group G is {x^0, x^1, ..., x^(n-1)} If there is a subgroup...
  31. A

    Lagrange theorem and subgroup help

    Homework Statement Let G be group, H<G , K<G, if gcd(lHl,lKl)=1, prove that H\bigcapK={1} Homework Equations The Attempt at a Solution so Lagrange theorem says that lHl l lGl, lKl l lGl, and of course 1 is inside both H and K, but how when they are coprime, the element are all...
  32. R

    What is the defining property of a subgroup in group G with elements A^g?

    Homework Statement Let G be a group and let A \leq G be a subgroup. If g \in G, then A^g \subseteq G is defined as A^g = \{ a^g | a \in A \} where a^g = g^{-1}ag \in G Show that Ag is a subgroup of G. The Attempt at a Solution I will use the one step subgroup test. First I have...
  33. M

    How to find the orbits of a subgroup?

    Homework Statement Hi everyone. I have just joined the community, and I really appreciate your help. Here is what I'm struggling with: Assume a permutation group G generated by set S, i.e., G=<S>. Since S is given, we can easily find the orbit partition for G. Now assume the subgroup H of G...
  34. K

    Is direct product of subgroups a subgroup?

    Hi.. In the second paragraph of the following paper, there is a statement: "Because the direct product of subgroups is automatically a subgroup.." http://jmp.aip.org/jmapaq/v23/i10/p1747_s1?bypassSSO=1 I don't see how that can be true...you can always take direct product of a subgroup...
  35. R

    Group theory, subgroup question

    Let A be a subgroup of G. If g \in G, prove that the set {g^{-1} ag ; a \in A} is also a subgroup of G. Thanks for any help.
  36. E

    Closure of a one-parameter subgroup

    I'm reading about a theorem that has as an assumption that the closure of some one-parameter subgroup is a torus. Could someone provide an example of a case where the closure of a one-parameter subgroup is of dimension greater than 1? Thanks.
  37. J

    Largest Cyclic Subgroup of S_n

    I became interested in this question a few weeks ago, I couldn't find much on it basically I've realized it's equivalent to finding for each n a partition of n say x_1,x_2,...,x_k such that x_1+x_2+...+x_k=n and lcm(x_1,...,x_k) is maximum (because you can then take the subgroup...
  38. K

    Symmetry breaking: what is the subgroup left?

    Hi... I have studied the standard model and know that spontaneous symmetry breaking by a vev breaks SU(2)xU(1) to a U(1). How do we know to what group a vev will break the original group? I have heard of Dynkin diagrams. Are they only for continuous groups? Is there any other method for...
  39. S

    Subgroup of given order of an Abelian group

    Hey! We know that if there exists an element of a given order in a group, there also exists a cyclic subgroup of that order. What about converse? Suppose there is a subgroup of an Abelian group of order 'm'. Does that imply there also exists an element of order 'm' in the Group. It does not...
  40. T

    Proving the Maximal Property of p-Sylow Subgroups in Finite Groups

    Homework Statement Suppose K is a normal subgroup of a finite group G and S is a p-Sylow subgroup of G. Prove that K intersect S is a p-Sylow subgroup of K. So I know that K is a unique p-sylow group by definition, is that enough to prove that the intersection of K with S is a p-sylow...
  41. L

    Reducing Normal Subgroup Calcs: Finite Groups Only?

    My abstract algebra book is talking about reducing the calculations involved in determining whether a subgroup is normal. It says: If N is a subgroup of a group G, then N is normal iff for all g in G, gN(g^-1) [the conjugate of N by g] = N. If one has a set of generators for N, it suffices...
  42. T

    Correspondence Theorem for Normal Subgroups in Groups of Order 168

    Homework Statement Show that if G is a group of order 168 that has a normal subgroup of order 4, then G has a normal subgroup of order 28. Homework Equations The Attempt at a Solution Let H be a normal subgroup of order 4. Then |G/H|=42=2*3*7, so then G?N has a unique, and...
  43. N

    Proving M is a Subgroup of G: Tips & Help

    Given a group G of order 22 and M = \{x \in G | x^{11}=e\}. Prove that M is a normal subgroup of G.I have troubles proving M is subgroup of G. If M was a subgroup, then I can show it is normal, but how to prove it's a subgroup? I know I have to show it's closed under multiplication and opposite...
  44. P

    Proved characteristic subgroup without using all facts

    Homework Statement This is question 30, section 2.5 from "Abstract Algebra 3rd edition" by Herstein. 2. Relevant information A subgroup H of group G is called characteristic if for all automorphisms phi of G, phi(H) is a subset of H. (I paraphrased this from question 29. I don't know...
  45. T

    Prove Z(G) Intersect H is Contained in Z(H) & Subgroup of H

    Homework Statement Let H be a subgroup of G. Prove Z(G) intersect H is contained in Z(H), use this result to verify that Z(G) intersect H is a normal subgroup of H. Given an example where Z(G) intersect H is a proper subgroup of Z(H). Homework Equations Z(G)= the center of G (x...
  46. M

    What is the Quotient Group S/S1?

    Does anybody know a general method to find the Group G/H (Where G is a Group and H is a subgroup of G) For example (1) What is the group S3/H ? S3 = {e, a, a^2, b, ab, (a^2)b} (Permutation group of order 6) H =< a >= {e, a, a^2} is a cyclic subgroup of G (2) What is the group...
  47. T

    Prove Intersection of Subgroups of G is Normal Subgroup

    Homework Statement Suppose that a group G has a subgroup of order n. Prove that the intersection of all subgroups of G of order n is a normal subgroup of G. Homework Equations The Attempt at a Solution I know that I need to do the following: Let S be the set of all subgroups of...
  48. T

    A4 Subgroup Question: Why is it Normal?

    Homework Statement There is only one subgroup of order 4 in A4 (Alternating group of degree 4)(This subgroup is (1), (12)(34), (13)(24), (14)(23)). Why does this imply that this subgroup must be a normal subgroup in A4? Generalize to arbitrary finite groups. Homework Equations...
  49. X

    Subgroup of external direct product

    I am trying to do the followin 2 problems but not sure if I am doing them correct. Anyone please have a look... 1. In Z40⊕Z30, find two subgroups of order 12. since 12 is the least common multiple of 4 and 3, and 12 is also least common multiple of 4 and 6. take 10 in Z40, and 10...
  50. G

    How to find the elements of a subgroup?

    If you have a subgroup and it's order how do you find the elements of the group? I'd be happy with any example to help explain this, but just so there's something to go off of my example would be if you have a subgroup A4 = <(1 2 3), (1 2)(3 4)> of S4 that has an order of 12 how do you find...
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