Subgroup Definition and 276 Threads

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...

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...

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...

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?

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!

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...

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...

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...

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

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...

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...

not homework question and i say yes.

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.

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

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.

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...

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?

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...

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?

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

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...

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?

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...

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...

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...

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...

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...

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.

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.

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

G,H be groups(finite or infinite) Prove that if (G:H)=n, then there exist some normal subgroup K of G (G:K)≤n! example) let G=A5, H=A4 then (G:H)=5, then K={id} exists, (G:K)≤5!

Homework Statement i just don't really get the one step subgroup test, which is very important, and something i should understand. can someone walk me through in general how to use the test? maybe give me a simple example? thanks. Homework Equations The Attempt at a Solution

I am somewhat distracted so this post will not be what it should, given that GAP is one of my interests. For those who don't already know: GAP is a powerful open source software package for computational algebra, especially computational group theory and allied subjects. This long running...

Homework Statement Let R = {all real numbers}. Then <R,+> is a group. (+ is regular addition) Let H = {a|a \epsilon R and a2 is rational}. Is H closed with respect to the operation? Is H closed with respect to the inverse? Is H a subgroup of G? Homework Equations N/A The Attempt at a...

Homework Statement Is the Cyclic Subgroup { (1), (123), (132)} normal in A_{4} (alternating group of 4) Homework Equations The Attempt at a Solution So I believe if I just check if gH=Hg for all g in A_4 that would be suffice to show that it is a normal subgroup, but that seems...

Homework Statement If H is a subgroup of G, show that g^{-1}Hg={g^{-1}hg \; h\in H is a subgroup for each g\in G Homework Equations The Attempt at a Solution I know I just have to check for closure and inverses, but the elements in this group g^{-1}hg with different h or with...

Where n >= 2. Is this true or false? I only got so far: If K is a subgroup of index 2, then it's normal. K is normal in Sn, so it's a union of conjugacy classes. Also, since |An K| = |An| |K| / |An intersection K| = 1/2n! * 1/2n! / |An intersection K| <= n!, then 1/4n! <= |An intersection...

It is said on wiki* that "Maximal compact subgroups are not unique unless the group G is a semidirect product of a compact group and a contractible group, but they are unique up to conjugation, meaning that given two maximal compact subgroups K and L, there is an element g in G such that...

Homework Statement List the elements of the subgroup generated by the given subset: The subsets {4,6} of Z12 Homework Equations The Attempt at a Solution so the GCD of 4 and 12 is 4, 12/4 = 3 elements that <4> generates : {0, 4, 8} GCD of 6 and 12 is 6, 12/6 = 2 elements...