# Tagged Questions

A group is an algebraic structure consisting of a set of elements together with an operation that satisfies four conditions: closure, associativity, identity and invertibility. Group theory studies groups.

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### $\{g\in G\mid\alpha(g)=g^{-1}\}=\frac34|G|$, find an abelian subgroup of index 2

$G$ is a finite group, $\alpha$ is an automorphism of $G$ and $I=\{g\in G\mid\alpha(g)=g^{-1}\}$. If $|I|=\frac34|G|$, show that $G$ has an abelian subgroup of index 2. Related question I don't ...
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### $R$ be a commutative unital ring , let $R^{\times}$ be the group of units of $R$ , then is it true that $(R,+) \ncong (R^{\times} , .)$?

Let $R$ be a commutative ring with unity , let $R^{\times}$ be the group of units of $R$ , then is it true that $(R,+)$ and $(R^{\times} ,\cdot)$ are not isomorphic as groups ? I know that the ...
Any permutation that moves $n$ elements in some fashion never revisiting the same until all others have been visited, in other words so that: $${\bf P}^n = {\bf I}, \text{ but no } 0<m<n \text{ ... 1answer 25 views ### Homomorphism between S4 and A4 I'm asked to find a group G with a subgroup H such that there is no normal subgroup N of G which performs: G/N =~ H. I thought of G=S4 and H=A4, because I don't think there is an homomorphism from S4 ... 1answer 57 views ### find torsion coefficients of groups I have to find torsion coefficients of groups G_1\simeq Z/2\oplus Z/4\oplus Z/3\oplus Z/3\oplus Z/9 and G_2\simeq Z/15\oplus Z/20\oplus Z/18. I want to ask if my calculations are correct. For ... 0answers 52 views ### If |G|=p^3q^2 then \Phi(G) is cyclic for primes p\neq q. I have conjectured this result for the Frattini subgroup by doing some calculations in GAP. I think this is even true if |G|=p_1^{i_1}\cdots p_n^{i_n} for i_j\leq 3 holds, but I would like to ... 1answer 21 views ### possible cyclic group from fundamental theorem of finite abelian Question: Give a representative of each Isomorphism class of Abelian group of order 225. Which ones are cyclic? By the Fundamental theorem of finite abelian group: \left | G \right |=225=3^{2}... 0answers 17 views ### Pronormal subgroups of direct products Suppose that G = A \times B. Let U = A \times \pi_B(U) \leq G such that \pi_B(U) is pronormal in B. Then U is pronormal in G. This is part of a proof of Proposition 4.3 in Pronormal ... 0answers 14 views ### Discrete subgroups of SU(m) \times SU(n) \times U(1) Is anything known about which permutation groups are subgroups of SU(m) \times SU(n) \times U(1)? Alternatively, how can one possibly go about finding them. I am trying to find discrete groups of ... 0answers 25 views ### Isomorphism between semidirect products Alperin and Bell, Groups and Representations, section 2, Proposition 11, p. 23, states: "Let H be a cyclic group and let N be an arbitrary group. If \varphi and \psi are monomorphisms from H ... 3answers 45 views ### Does the converse of Lagrange's theorem hold for any finite Dedekind group? I'm currently reading a book on algebra and the following argument is used to prove that the converse of Lagrange's theorem (if d divides |G| there exists a subgroup of order d) holds for any ... 3answers 57 views ### Is my proof True ? ( about Group theory, direct product ) I have a problem. It states that: Let G is a group and |G|=mn, (m,n)=1. Assume that G has exactly just one subgroup M with order m and one subgroup N with order n. Prove: G is ... 2answers 24 views ### Normal closure of a subgroup of a free group. Let G be a finitely generated free (nonabelian) group, H a subgroup generated by some of the generators of G, and a: G\to AG be the projection to the abelianization AG:=G/[G,G]. Is it true ... 0answers 32 views ### Question about generators and Hom functor In a given category \mathcal{C} I want to prove the following statement: If U es a generator in the category \mathcal{C} if and only if the left-exact functor Hom_{\mathcal{C}}(U,-): \mathcal{... 0answers 21 views ### How can I prove that the the number of elements of order k in \mathbb{Z}_n is φ(k)? [on hold] How can I prove that the number of elements of order k in \mathbb{Z}_n is ϕ(k) where k is number that divides n ? 4answers 57 views ### If G is a non-abelian group of order 10, prove that G has five elements of order 2. I'm trying to prove this statement: If G is a non-abelian group of order 10, prove that G has five elements of order 2. I know that if a\in G such that a\neq e, then as a ... 0answers 44 views ### Error Correcting Code and Graph Theory I am currently in an introductory graph theory class, and we are supposed to give a short presentation by the end of the semester. Recently, I've learned (a very small amount) about error correcting ... 4answers 618 views ### Problems from the Kourovka Notebook that undergraduate students can fully appreciate The Kourovka Notebook is a collection of open problems in Group Theory. My question is: could you point out some (a "big-list" of) problems [by referencing them] presented in this book that are, ... 4answers 53 views ### \mathbb{C}/\mathbb{Z} is isomorphic to multiplicative group \mathbb{C}\setminus\{0\} [duplicate] I have to show that \mathbb{C}/\mathbb{Z} is isomorphic to the multiplicative group \mathbb{C} \setminus \{0\}. Proof. Let f:\mathbb{C} \setminus \{0\} \rightarrow \mathbb{C}/\mathbb{Z} be the ... 2answers 50 views ### How can I find all the numbers of order 10 in Z_{60}? [on hold] How can I find all the numbers of order 10 in Z_{60}? In fact, these are all the numbers with (x,60)=6. How can I find all these numbers? 1answer 43 views ### When is (\Bbb Z/n\Bbb Z)^\times cyclic? [duplicate] Is the group of units (\Bbb Z/n\Bbb Z)^\times always cyclic? Do we need that n is a prime or something? 1answer 22 views ### All homomorphisms from Z/4Z to Z/6Z I am asked to find all group homomorphisms from Z/4Z to Z/6Z. Let f:Z/4Z \rightarrow Z/6Z be such a homomorphism. By definition we have f(1) = 1 and therefore f(0)=f(1 * 0) = f(1) * f(0) = ... 2answers 19 views ### order of an element in a modulo group under multiplication Suppose G is the group ℤ_{37}^\times under multiplication. Then is there a way that I can prove the order of the element 2 in G is 36 without finding all the powers of 2 until I get unity?... 1answer 89 views ### Finding permutation with a condition Let a be a permutation in S_6. I'm asked whether there is an a so that a^2 = (123)(456) I'm quite confused about where to start. I do know the a must consist of 3 elements (right?). How ... 2answers 178 views ### About an article regarding free groups I am currently reading this paper https://blms.oxfordjournals.org/content/35/5/624.abstract and I have some difficulties to understand two steps of the proof of the main theorem. Let G be a non ... 3answers 16 views ### Sylow p-subgroups in S_p and order p elements I realise this question has been asked a few times before, but I don't understand the answers. What is the number of Sylow p subgroups in S_p? Why are the order p elements in S_p exactly cycles ... 0answers 17 views ### Direct product of join of subgroups Suppose that G is a finite group with A, B, H, K \leq G. Suppose that H\times A \leq G and K\times B \leq G. I want to show that \langle H,K \rangle \times \langle A, B \rangle = \langle H \... 1answer 74 views ### What is the number of Sylow p subgroups in S_p? I am reading the Wikipedia article entitiled Sylow theorems. This short segment of the article reads: Part of Wilson's theorem states that (p-1)! is congruent to -1 (mod p) for every prime p. One ... 2answers 55 views ### Let G be a (probably infinite) group and H \leq K \leq G. |G:H| < \infty \Rightarrow |G:H|=|G:K||K:H|? Let G be a (probably infinite) group and H \leq K \leq G. Is it true that if we have |G:H| < \infty then |G:H|=|G:K||K:H| ? Thanks in advance. 1answer 25 views ### If K\leq H\leq G (not necessarily finite groups). Then prove that [G:K]=[G:H]\cdot [H:K] [duplicate] Let K\leq H\leq G (not necessarily finite groups). Why do we have [G:K]=[G:H]\cdot [H:K]? I can't figure out a proof in the setting of possibly infinite groups and non-normal subgroups. 2answers 94 views ### Let G be finite group and H<G. For every proper subgroup K, [G:H]\leq[H:K]. Let G be finite group and H<G. For every proper subgroup K, [G:H]\leq[H:K]. I want to prove H is normal subgroup. I fixed K:=g^{-1}Hg but this doesn't work. Can somebody advise me? 2answers 57 views ### Normal subgroups of A_5 must contain a 3-cycle. I am trying to prove the simplicity of A_5 by showing that every non-trivial normal subgroup H contains a 3-cycle, and therefore is all of A_5 since the 3-cycles all belong to one conjugacy ... 0answers 36 views ### Regarding the general method of the ''Classify groups of order X'' question. Anyone who has had to prepare for an algebra qualifying exam is familiar with the "Classify groups of order X" question. To illustrate my general question, which I postpone until the end, consider ... 2answers 47 views ### Non-abelian Order of 6 is isomorphic to S_3 [on hold] I know that it's duplicate, but , How can I prove it? I know that must be element "a" of order 2, and element "b" of order 3. What is the next step? In fact, from what I search it is claimed that ... 0answers 30 views ### Can someone find example for it? [on hold] Example for group that for any element the order is 2 apart of e. (without Z_2+Z_2+Z_2+\cdots+Z_2) 0answers 25 views ### It's True? For any group G that his order is pq {p,q primes} has 2 nontrivivallic normal subgroups? [on hold] It's True? For any group G that his order is pq {p,q primes} has 2 nontrivivallic normal subgroups? If yes, so hwo can i prove it? [without Sylow Theorems] p,q > 1 1answer 160 views ### Does this particular axiom on a semigroup guarantee that it is a group? Update: Eric Wofsey has demonstrated the conjecture in the commutative case below, and Tobias Kildetoft has provided a simple counterexample to the non-commutative claim. This would-be replacement ... 2answers 33 views ### G a finite group with two non-trivial normal subgroups. |G| = pq. Why G cycle? How can I prove that if G is a finite group , and the order of G is pq while p and q are primes, and in addition , G with two normalic subgroups , so --> G is cycle? Ideas? Hwo can i ... 0answers 34 views ### Automorphism group of finite p-groups firstly I apologize for my naive knowledge in group theory. Let G be a finite p-group with automorphism group {\rm Aut}(G) and let N be a maximal subgroup of G. Let g be ... 1answer 26 views ### Linear representations of projective groups Does the projective linear group PSL_2(\mathbb{R}) admit faithful linear representations? In other words, does there there exist a homomorphism SL_2(\mathbb{R}) \to GL_n(\mathbb{R}), for some n, ... 1answer 36 views ### How to explain this contradiction about Weyl group of SL_n(K)? I have some difficulties in understanding why the Weyl group of algebraic group SL_n(K) is isomorphic to symmetric group S_n. Let G=SL_n(K) be the simply-connected algebraic group over the ... 1answer 417 views ### Proving that a group of order pqr (with conditions on those primes) is abelian. I'm doing this exercise: Let G be a group, with |G|=pqr, p,q,r different primes, q<r, r \not\equiv 1 (mod q), qr<p. Also suppose that p \not\equiv 1 (mod r), p \not\equiv ... 2answers 37 views ### Questions about Sylow p-groups Question 1 Is it true that there is only one Sylow p-group in an abelian group? Question 2 If there is only one Sylow p-group, then it is normal, true? Because if H\leq G is a Sylow p-group ... 1answer 9 views ### number of orbits by action of D_{12} on \mathbb{Z}_{12}^k Let X=\mathbb{Z}_{12}^k for k\in \mathbb{N} and G=D_{12}. Define an action of D_{12} on X by setting rotations r^n(p)=(p_1+n,\dotsc,p_k+n) where the coordinates are taken modulo 12 and ... 0answers 26 views ### Is power-associativity an equational property? A magma M is said to be power-associative if the subalgebra generated by any element is associative. This can be written simply as x^mx^n=x^{m+n} for all m,n positive integers and x\in M, ... 2answers 26 views ### Trying to find an isomorphism I'm trying to find an isomorphism from Z31 to itself that maps 13 to 28.I used the extended euclid algorithm to find that 12 is the inverse of 13, but that's as far as I got. What do I do next? How ... 6answers 8k views ### Order of elements in abelian groups How can I prove that if G is an Abelian group with elements a and b with orders m and n, respectively, then G contains an element whose order is the least common multiple of m and n? ... 2answers 803 views ### Abelian Group Element Orders [duplicate] I want to show that if a finite abelian group has elements of order m and n then it will have an element of order \text{lcm}(m,n). First I proved the lemma if a has order m and b has ... 3answers 389 views ### A group such that a^m b^m = b^m a^m and a^n b^n = b^n a^n (m, n coprime) is abelian? Let (G,.) be a group and m,n \in\mathbb Z such that \gcd(m,n)=1. Assume that$$ \forall a,b \in G, \,a^mb^m=b^ma^m,\forall a,b \in G, \, a^nb^n=b^na^n. Then how prove $G$ is an abelian ...
I noted a discussion on groups being abelian under a certain restriction on powers of elements, e.g. http://tiny.cc/chs45. Maybe this result (probably not too well-known) concludes it all. Let $m$ ...