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# Group Theory

### Prove that a group of order 108 must have a normal subgroup

Prove that a group of order 108 must have a normal subgroup of order 9 or 27.

### Group Theory Proof

See the attached file. Modern Algebra Group Theory (CXII) Groups of Order Power of a Prime Another Counting Principle

### Permutation Groups

See the attached file. Modern Algebra Group Theory (CXI) Permutation Groups Another Counting Principle If O(G) = pn a prime

### Sylow's Theorem Group Theory

Modern Algebra Group Theory (CX) Sylow's Theorem Find all 3-Sylow subgroups of or, Sylow 3-subgroups and 2-Sylow subgroup or, Sylow 2-subgroups of the symmetric group of degree 4, S4.

### Sylow's Theorem Symmetric Group

Modern Algebra Group Theory (CIX) Sylow's Theorem In the symmetric group of degree 4, S4 , find a 2-Sylow subgroup and a 3-Sylow subgroup.

### Group Theory : Permutaion Groups and Counting Principles

Using O(N) = &#931;ca for some choices of a in N , prove that in A5 there is no normal subgroup N other than (e) and A5 .

### Group Theory : Prove that O(N) = &#931; ca for some choices of a in N.

Prove that O(N) = &#931; ca for some choices of a in N.

### Group Theory: Find all the Conjugate Classes

Find all the conjugate classes in A5 and the number of elements in each conjugate class.

### Group Theory :finite group G an element exactly two conjugate

If in a finite group G an element a has exactly two conjugates, prove that G has a normal subgroup N &#8800; e , G

### Group Theory: Number Of Conjugates

Find the number of conjugates of (1 2)(3 4) in Sn , n &#8805; 4. Please see the attachment for proper formatting.

### Group Theory: Prove that any element &#963; in Sn which commutes with (1 , 2 , ... , r) is of the form &#963; = (1 , 2 , ... , r)^i &#964; where i = 0, 1 , 2 , ... , r , &#964; is a permutation leaving all of 1 , 2 , ... , r fixed.

Prove that any element &#963; in Sn which commutes with (1 , 2 , ... , r) is of the form &#963; = (1 , 2 , ... , r)^i &#964; where i = 0, 1 , 2 , ... , r , &#964; is a permutation leaving all of 1 , 2 , ... , r fixed.

### Group Theory : Conjugates and Permutation Groups

Find the number of conjugates that the r-cycle (1 , 2 , ... , r) has in Sn .

### Group Theory : Permutation Groups

Given the permutation x = ( 1 2 ), y = ( 3 4 ) find a permutation a such that a^( - 1) x a = y.

### Group Theory : Cycles and Permutation Groups

In Sn prove that there are (1/r). [n!/( n - r ) ] distinct r cycles.

### Group Theory: Permutation Groups

Given the permutation x = ( 1 2 )( 3 4 )( 5 6 ), y = ( 1 3 )( 2 5 )( 4 6 ) find a permutation a such that a^( - 1) x a = y.

### Group Theory: Permutation Groups

Given the permutation x=(1 2)(3 4), y=(1 4)(2 3), find a permutation a such that a^(-1)xa=y. Thank you for the help!

### Group Theory : Permutation Groups

Prove that the smallest subgroup of Sn containing ( 1 , 2 )and ( 1 , 2 , ... , n ) is Sn. ( In other words, these generate Sn )

### Group Theory : Permutation Groups and Even Permutations

Determine which of the following are even permutations: (a) ( 1 2 3 ) (b) ( 1 2 3 4 5 )( 1 2 3 )( 4 5 ) (c) ( 1 2 )( 1 3 )( 1 4 )( 2 5 )

### Continuous Maps, Homomorphisms and Cyclic Groups

Let f: S^n --> S^n be a continuous map. Consider the induced homomorphism f*: H~_n (S^n) --> H~_n (S^n), where H~_n is a reduced homology group. Then from the fact that H~_n (S^n) is an infinite cyclic group, it follows that there is a unique integer d such that f*(u) = du for any u in H~_n (S^n). How exactly does "ther

### Topological Groups : Quotient and Open Maps

Let be p:G to G/N a quotient map.Is it open? Let be f:G to H an open map. Is it quotient map? Here G and H are topological groups, and N is an subgroup.

### Group Theory : Permutation Groups

Prove that there is no a such that a^( - 1 ) ( 1 , 2 , 3 ) a = ( 1 , 3 )( 5 , 7 , 8 )

### Group Theory : Permutation Groups

Compute a^( - 1 )ba where a = ( 5 7 9 ) b = ( 1 2 3 )

### Group Theory :The order of an n-cycle, the order of the product of the disjoint cycles and order of a given permutation.

Modern Algebra Group Theory (LXXXIV) Permutation Groups

### Group Theory: Inverses of Cycles and Permutuations

Modern Algebra Group Theory (LXXXII) Permutation Groups The Invers

### Normal Subgroup of Order

Modern Algebra Group Theory (LXXV) Normal subgroup of a group The group of order p^2, where p is a prime number Prove that a group of orde

### Nilpotent Groups

Let G = UT (n,F) be the set of the upper triangular n x n matrices with entries in a field F with p elements and 1's on the diagonal. The operation in G is matrix multiplication. (a) Show that G is a group (b) Show that G is a finite p-group (c) Consider the upper central series of G: 1 = Z_0 (G) <= Z_1 (G) <= Z_2 (G) <

### Group of order 9

This is the question: Consider small groups. (i) Show that a group of order 9 is isomorphic to Z9 or Z3 x Z3 (ii) List all groups of order at most 10 (up to isomorphism)

### Disjoint cycles and least common multiple

Symmetric groups: G = Sn. (i) Let g1, g2 belong G be two disjoint cycles, and let g = g1g2. Prove that o(g) = lcm { o( g1), o(g2)}, where lcm stands for the least common multiple. (ii) Let g= g1g2 ... gr belong G, where g1,g2, ... gr are disjoint cycles. Prove that o(g) = lcm {o(g1), o(g2), ... o(gr)}. Can you tell me

### Group Theory : Let G be a group and Z(G), the centre of G, then G/Z(G) is equivalent to I(G) , where I(G) is the set of all inner automorphisms of G.

Modern Algebra Group Theory (LXXII) The Set of all Automorphisms of a Group The Set of all Inner Automorphisms of a Group

### Path-connected Space : Abelian Group

Let x0 and x1 be points of the path-connected space X. Show that Pi_1(X,x0) is abelian iff for every pair a and b of paths from x0 to x1, we have a'=b', where a'([f])=[a-]*[f]*[a];( a- means the reverse of a.) and [f] belongs to Pi_1(X,x0). a':Pi_1(X,x0)->Pi_1(X,x1).