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# Group Theory

### Group Theory : Cayley Tables, Cyclic Groups and Isomorphisms

? I have the following cayley tables (which is in modulo 9) determine the order of each element . Prove that G is a cyclic group. ? Let be the symmetric group of degree 3 together with composition of maps. Is G isomorphic to ? Justify your answer. ? Let p be a prime number and G a group of order with identity e

### Groups and Fields

Find H K in {see attachment}, if H={,}, and K = {, , , 13], ,}.

### Direct Product Groups

Let G1 and G2 be groups, and let G be a direct product of G1 x G2. Let H = {(x1, x2) element G1 x G2 | x2 = e} and let K = {(x1, x2) element G1 x G2| x1 = e}. (a) Show that H and K are subgroups of G. (b) Show that HK = KH = G (c) Show that H [see attachment] K = {(e,e)}.

### Permutation Groups : Order of Rigid Motions of a Cube

Find the order of the group of rigid motions of a cube.

### Groups and Subgroups

Let G sub1 and G sub 2 be groups, with subgroups H sub 1 and H sub 2, respectivetly. Show that {(x sub 1, x sub 2) | H sub 1 is an element of H sub 1, x sub 2 is an element of H sub 2} is a subgroup of the direct product G sub 1 x G sub 2.

### Abelian Groups

Prove that if G1 and G2 are abelian groups, then the direct product G1 x G2 is abelian.

### Groups: Third Sylow Theorem

Suppose a simple group G of order 660 is a subgroup of the symmetric group S11 (here S is with 11 as a subscript ) and x = {1,2,3,4,5,6,7,8,9,10,11}( a permutation like 1 goes to 2 and so on, I think ) in G. If P equals the span of x ( or <x> ) determine the permutations which generate NGP ( here G is written as the subscript of

### Rings and Groups (Ring Theory, Quaternions, Homomorphisms, Matrices)

A number of questions involving rings and groups. Example: 3) Let R be a ring and [equationA]. Let [equationB] be the ring of n x n matrices with entries in R. What is the identity element of S? *(Please see attachment for complete list of problems)

### Proofs in Group Theory: Cayley Table, Subsets and Cosets

Five Problems: Let G=[FORMULA1], with operation given by multiplication modulo 14. 1) by computing the Cayley table of G, or otherwise, show the G is a group. You may assume that without proof multiplication modulo 14 is associative. 2) Prove that the subset H={1,9,11} is a subgroup of G 3) Compute the left cosets of G a

### Definea Group using Axioms

Define * on Z by a * b = max{a,b}. Is Z a group?

### Abelian Groups Distinct Elements

Show that a abelian group must have five distinct elements

### Group Theory for Symmetric Groups

1) Recall that So denotes the symmetric group of degree 6, the group of permutations f the numbers 1 to 6 ... see attached for full questions 1) Recall that denotes the symmetric group of degree 6, the group of permutations f the numbers 1 to 6. let . Thus is a bijection, mapping 1 to 3, 2 to 5, etc. let • C

### Group Theory: Matrix Manipulation and Abelian Group

1) Let G be the set of 2x2 matrices given by show that G is a group under matrix multiplication. You may assume that matrix multiplication is associative. 2) Let G be a group,... prove that G is Abelian. See attachment.

### Cryptology Square Functions

Suppose that the ADFGVX square is: _ A D F G V X A F L 1 A 0 2 D J D W 3 G U F C I Y B 4 P G R 5 Q 8 V E V 6 K 7 Z M X X S N H O T 9 and that they keyword is : GERMAN Decipher: AFAXA XAFGF XDDAV DAGGX FGXDD XVVAV VGDDV FADVF ADXXX AXA

### Groups : Symmetry

Note: G' means derived (commutator) subgroup of G and Sn is symmetric group of degree n Please find G' in each case (a) G is abelian (b) G = Sn

### Groups and Subgroups : Indicies

If K is a normal subgroup of G has index m, show that g^m belongs to K for all g belonging to G.

### Groups and elements

If G is a group of order p^k, where p is a prime and k >=, show that G must have an element of order p.

### Abelian group

If G is any group, define \$:G->G by \$(g) = g^-1. Show that G is abelian if an only if \$ is a homomorphism.

### Cyclic Groups

Show that every cyclic group Cn of order n is abelian. (Moreover, show that if G is a group, so is GxG)

### Infinite Group Elements

Find an infinite group such that every element has finite order.

### Abelian Groups Examples

Find examples of the following. Explain your answers. (a) A nonabelian group G and a proper normal subgroup S such that G/S is cyclic.

### Modern/Abstract Algebra Problem (Group Theory)

What are the normal subgroups in S_4? I would appreciate it if someone could also give a thourough explanation of how they came up with the answer.

### Modern/Abstract Algebra Problem (Group Theory)

What is the number of different conjugated classes of elements of order 2 in S_5.

### Subgroups of cyclic groups

Show that any subgroup of a cyclic group is cyclic.

### Find the mean of the group data

Using the employment information in the table on Alpha Corporation, find the mean for the grouped data. Years of service Frequency 1-5 5 6-10 20 11-15 25 16-20 10 21-25 5 26-30 3

### Group homomorphisms functions

Homomorphism Problem 4: Let G, G1, and G2 be groups. Let µ1 : G -> G1 and µ2 : G -> G2 be group homomorphisms. Prove that µ : G -> G1 × G2 defined by : µ (x) = (µ1 (x), µ2 (x)), for all x in G, is a well-defined group homomorphism.