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

    Proof of union of subgroups as a subgroup

    Prove that if (H,o) and (K,o) are subgroups of a group (G,o), then (H \cap K,o) is a subgroup of (G,o). Proof: The identity e of G is in H and K, so e \in H\capK and H\capK is not empty. Assume j,k \in H\capK. Thus jk^{-1} is in H and K, since j and k are in H and K. Therefore, jk^{-1}...
  2. S

    Proving a Subgroup of a Solvable Group is Solvable

    I'm supposed to show that a subgroup of a solvable group is solvable. (I am using the Fraleigh Abstract Algebra book and the given definition of a solvable group is a group which has a COMPOSITION series in which each of the factor groups is abelian. In other books I have looked at a solvable...
  3. E

    Characteristic Subgroup is Normal

    Homework Statement Recall that a subgroup N of a group G is called characteristic if f(N) = N for all automorphisms f of G. If N is a characteristic subgroup of G, show that N is a normal subgroup of G. The attempt at a solution I must show that if g is in G, then gN = Ng. Let n be in N...
  4. N

    What Is the Smallest Normal Subgroup of a Group Containing a Given Subset?

    1) Let X be anon empty subset of a group G .prove that there is a smallest normal subgroup of G containing X ii)what do we call the smallest normal subgroup of G containing X
  5. P

    Isomorphism of D5 and S5: Finding Elements and Permutations

    Homework Statement By considering the vertices of the pentagon, show that D5 is isomorphic to a subgroup of S5. Write all permutations corresponding to the elements of D5 under this isomorphism. The Attempt at a Solution To show isomorphic, need to find a function f: D5->S5, where...
  6. P

    Are Even Permutations a Subgroup of D4?

    Homework Statement Consider the group D4 (rigid motions of a square) as a subgroup of S4 by using permutations of vertices. Identify all the even permutations and show that they form a subgroup of D4. The Attempt at a Solution I think I have the permutations of correct. They are...
  7. P

    Is H1xH2 a Subgroup of G1 X G2?

    Homework Statement Let G1, G2 be groups with subgroups H1,H2. Show that [{x1,x2) | x1 element of H1, x2 element of H2} is a subgroup of the direct product of G1 X G2 The Attempt at a Solution I'm not sure how to begin solving this problem.
  8. H

    What is the cyclic subgroup order of GL(2,p^n) generated by the given matrix?

    I'm trying to prove that GL(2,p^n) has a cyclic subgroup of order p^{2n} - 1. This should be generated by \left( \begin{array}{cc} 0 & 1 \\ -\lambda & -\mu \end{array} \right) where X^2 + \mu X + \lambda is a polynomial over F_{p^n} such that one of its roots has multiplicative order...
  9. N

    If K is a subgroup of G of order p^k, show that K is subgroup of H

    Homework Statement Let |G| = (p^n)m where p is prime and gcd(p,m) = 1. Suppose that H is a normal subgroup of G of order p^n. If K is a subgroup of G of order p^k, show that K is subgroup of H. Homework Equations The Attempt at a Solution Okay, I wonder if there is more I need...
  10. M

    Conjugates in the normalizer of a p-Sylow subgroup

    [SOLVED] Conjugates in the normalizer of a p-Sylow subgroup Homework Statement Let P be a p-Sylow subgroup of G and suppose that a,b lie in Z(P), the center of P, and that a, b are conjugate in G. Prove that they are conjugate in N(P), the normalizer of P (also called stablilizer in other...
  11. P

    Is G isomorphic to a subgroup of G/M\times G/N?

    Homework Statement Let M and N be normal subgroups of G, and suppose that the identity is the only element in both M and N. Prove that G is isomorphic to a subgroup of the product G/M\times G/N Homework Equations Up until now, we've dealt with isomorphism, homomorphisms, automorphisms...
  12. I

    Normalizer &amp; normal subgroup related

    [SOLVED] Normalizer &amp; normal subgroup related Let H \subset G. Why is H a normal subgroup of its own normalizer in G?
  13. P

    Inner Automorphisms as a Normal Subgroup

    Homework Statement Let G be a group. We showed in class that the permutations of G which send products to products form a subgroup Aut(G) inside all the permutations. Furthermore, the mappings of the form \sigma_b(g)=bgb^{-1} form a subgroup inside Aut(G) called the inner automorphisms and...
  14. E

    Commutator subgroup and center

    Homework Statement Please confirm that the center of a group always contains the commutator subgroup. I am pretty sure its true. Homework Equations The Attempt at a Solution
  15. E

    What Defines the Smallest Normal Subgroup Containing a Subset?

    [SOLVED] smallest normal subgroup Homework Statement Given any subset S of a group G, show that it makes sense to speak of the smallest normal subgroup that contains S. Hint: Use the fact that an intersection of normal subgroups of a group G is again a normal subgroup of G.Homework Equations...
  16. L

    Isomorphic Subgroup Action on a Group: Proving the Isomorphism between H^g and H

    Homework Statement I'm trying to prove that, if H is a subgroup of an arbitrary group G, then H^g, the action of a given element in G on H, is isomorphic to H. Homework Equations The Attempt at a Solution Let \sigma denote a given action of G on H. We are considering the map...
  17. B

    Every finite subgroup of isometries of n-dimensional space

    ... fixes at least one point. I recently came upon the proof in a book and I didn't quite understand the notion of "rigid motion", and I was wondering if you could help clarify it for me. Is it just "the vertices must stay in the given order", as used in symmetries of polygons? I've attached...
  18. A

    Normal subgroup of prime order in the center

    Homework Statement Let H be a normal subgroup of prime order p in a finite group G. Suppose that p is the smallest prime dividing |G|. Prove that H is in the center Z(G). Homework Equations the Class Equation? Sylow theorems are in the next section, so presumably this is to be done without...
  19. E

    Proving a nonempty finite subset of a group is a subgroup

    Homework Statement Let a non empty finite subset H of a group G be closed under the binary operation * on G. Show that H is a subgroup of H. 2. Relevant Definitions Group Properties: G1: a*(b*c)=(a*b)*c for all a,b,c in G G2: e*x=x*e=x for all x in G G2: if x is in G then x'...
  20. B

    Showing that a subgroup is subnormal in the original group G

    Question: Let G be a group of order p^n > 1 where p is prime. If H is a subgroup of G, show that it is subnormal in G. That is, I need to show that there is a chain of subgroups H=H_0 \triangleleft H_1 \triangleleft ...\triangleleft H_m = G, where m\leq n. Analysis: We can easily show by...
  21. M

    An ideal in a ring as 'analogous' to a normal subgroup of a group, but

    Hello all. I am in need of a quick clarification. A text I am reading describes an ideal in a ring as 'analogous' to a normal subgroup of a group but there appears to be a slight difference in structure in that a member of the underlying additive group from which the ideal is formed operates...
  22. U

    Understanding Finite Abelian Subgroups of GL(n, C)

    How do finite abelian subgroups of GL(n, C) with n > 1 look like ? I would say the elements of those subgroups are only the diagonal matrixes but I am not sure (for my homework I do not have to prove it but I want to use this result if it is true). GL(n, C) are all the invertible matrixes over...
  23. happyg1

    What are the Frattini subgroups of D2n and D∞?

    Homework Statement Find Frat(D_{2n}),Frat(D_{\infty})Homework Equations Frat is the set of all nongenerators of a group. The Attempt at a Solution I know that D_2n is generated by a rotation of 360/n with order n, and a reflection, f of order 2. So D_2n=<r,f> and any element that can be...
  24. quantumdude

    5-Sylow Subgroup of Groups of Order 90.

    Homework Statement Show that the 5-Sylow subgroups of a group of order 90 is normal. Homework Equations None. The Attempt at a Solution I know that the number \nu_5 of 5-Sylow subgroups must divide 90 and be congruent to 1 mod 5. That means that \nu_5\in\{1,6\}. I also know that...
  25. V

    How can the normality of a subgroup be proven when the group has a finite index?

    Having trouble with a couple of algebra questions and would really appreciate any hints or pointers. 1. A is a subgroup of group G with a finite index. Show that N = \bigcap_{x \in G}x^{-1}Ax is a normal subgroup of finite index in G. I'm able to show that N is a subgroup of G by applying the...
  26. quantumdude

    Subset of the Group of Permutations: Subgroup or Not?

    Well, in 5 years of PF'ing and watching over this forum, I am finally posting my first homework question. :-p I'm taking a graduate course in Algebra, and it's been 11 years since I took the undergraduate version. So, I'm going back and doing all the homework exercises in my undergrad book...
  27. R

    Is H a Subgroup of G Given Specific Conditions?

    Homework Statement Let H be a subset of a group G. Prove: H does not equal the empty set and a,b are contained in H, which implies ab^(-1) is contained in H, which implies H is a subgroup. Let G be a group. Let Ha = {x is contained in G | ax = xa }. Prove Ha is a subgroup of G ...
  28. quasar987

    What Defines a Normal Subgroup in Group Theory?

    My professor of topology gave us a quick overview of the group theory results we will be needing later and among the things he said, is that a normal subgroup of a group G is a subgroup H such that for all x in G, xHx^{-1}=H. Is this correct? The wiki article seems to indicate that equality...
  29. A

    Characters of Normal Subgroup of Index 2

    Hi There, Ok, I'm new to this so I'm sorry if this is abit warbled!... We have a normal subgroup N of a finite group H such that [H:N]=2 We have a chatacter Chi belonging to the irreducible characters of H, Irr(H) , which is zero on H\N. I have already shown that Chi restricted to N =...
  30. E

    How Do I Verify Identity and Closure in a Matrix Subgroup?

    Subgroup help please! Urgent! Hi, Can anyone help me with the following question? Many thanks! Again I'm working from a book so I'm must trying to apply theory to exercises -__- http://i12.tinypic.com/3zsyuf8.jpg How do I show that the identity lies in this? And also show that we...
  31. E

    Find Subgroup of Order 4 in $\mathbb{Z}$ /13

    The question reads as: "Let G = ($\mathbb{Z}$ /13)^*. Find a subgroup H of G such that |H| = 4. " I think this means that you have to find a subgroup that has order 4. Although I'm not entirely sure what that means in this context. Any help will be appreciated.
  32. mattmns

    Proving SL(2,R) is a Subgroup of GL(2,R)

    I have a problem that states Define the Special linear group by: (Let R denote real numbers) SL(2,R) = \{ A\in GL(2,R): det(A)=1\} Prove that SL(2,R) is a subgroup of GL(2,R). ___ Now a subset H of a group G is a subgroup if: i) 1 \in H ii) if x,y \in H, then xy \in H iii) if...
  33. H

    Groups containing no proper subgroup

    1. Describe all groups G which contain no proper subgroup. This is my answer so far: Let G be a such a group with order n. Then the following describe G: (a) Claim that every element in G must also have order n. Proof of this: If this wasn't true, the elements of lower order (elements of...
  34. A

    Are Sylow subgroups unique in groups with specific conditions?

    Suppose H and K are subgroups of G with H normal in K, |H||K| = |G|, and the intersection of H and K being identity. Then HK = G. Since HK is the union of hK for all h in H and since hK = h'K iff h = h', wouldn't the set of cosets of K be {hK : h in H}? Also, wouldn't this form a group...
  35. S

    Subgroup GH (Growth hormone releasing) pituitary adenomas by microarray

    Subgrouping expression profiles of GH pituitary adenomas Hello Everybody, I am going to subgroup GH (Growth hormone releasing) pituitary adenomas according to their expression profiles by microarray and RT-qPCR. I do have 50 GH pituitary adenomas to subgroup and i wonder should i run all...
  36. C

    Math Help for Finite Cyclic Group & Subgroups

    hey! great to find such an informative website... i'm an undergrad math student and have lots of problems with group theory.i hope you'll all help me enjoy group theory... my teacher put forward these question last week and I've been breaking my head over them without much success : 1. let G...
  37. D

    Proving 1 p-Sylow Subgroup of G is Normal

    Hello! For the life of me, I can't seem to figure this out (vapor lock in the ol' brain): Show that if G has only 1 p-Sylow subgroup, then it must be normal. I know it something to do with showing it's a conjugate to itself (right coset = left coset?). I'm just not quite sure how to go...
  38. M

    Determine 3 Elements of Group H with Primes p & q

    let p and q be distinct primes. suppose that H is a proper subset of integers and H is a group under addition that contains exactly 3 elements of the set { p,p+q,pq, p^q , q^p}. Determine which of the foll are the 3 elements in H a. pq, p^q, q^p b. P+q, pq,p^q c. p, p+q, pq d. p...
  39. S

    Proving A4 Has No Subgroup of Order 6

    Prove that A4 has no subgroup of order 6 in this way (and this way only): Suppose that A4 has a subgroup H of order 6. Explain (in one sentence) why H must contain a 3-cycle. WLOG(without loss of generality) let this be (1,2,3). Then H must have iota,(1,2,3) and (1,2,3)^-1=(3,2,1). Now...
  40. J

    Proof of Normal Subgroup Thm- Alternative Strategies

    Can anyone come up with an alternative proof of the following? If H, a subgroup of G, has index [G:H]=p where p is the smallest prime dividing |G|, the H is normal in G. I'm already aware of one proof, given here http://www.math.rochester.edu/courses/236H/home/hw8sol.pdf (page 3 -...
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