Subgroup Definition and 291 Threads

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...

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...

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...

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

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...

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...

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...

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...

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...

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...

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?

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...

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...

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?

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!

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?

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 *.

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...

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?

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

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?

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...

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...

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...

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...

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...

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?

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...

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...

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...

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...

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|...

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...

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...

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...

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

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?

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...

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...

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...

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...

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.

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...

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...

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...

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?

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...

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...

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...

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...