Use the classification of groups with six elements to show that A(4) has no subgroup with...

Use the classification of groups with six elements to show that A(4) has no subgroup with 6 elements. [ Hint: check that the product of any two elements of A(4) of order 2 has order 2]

Expert Solution

Colby Messinger answered 2 years ago

Show that the groups of the following orders have a normal Sylow subgroup. (a) |G| =...

Show that the groups of the following orders have a normal Sylow subgroup. (a) |G| = pq where p and q are primes. (b) |G| = paq where p and q are primes and q < p. (c) |G| = 4p where p is a prime greater than four.

Show that M is a subgroup N; N is a subgroup D4, but that M is not a subgroup of D4

D4 = {(1),(1, 2, 3, 4),(1, 3)(2, 4),(1, 4, 3, 2),(1, 2)(3, 4),(1, 4)(2, 3),(2, 4),(1, 3)} M = {(1),(1, 4)(2, 3)} N = {(1),(1, 4)(2, 3),(1, 3)(2, 4),(1, 2)(3, 4)} Show that M is a subgroup N; N is a subgroup D4, but that M is not a subgroup of D4

Show that there are only two distinct groups with four elements, as follows. Call the elements...

Show that there are only two distinct groups with four elements, as follows. Call the elements of the group e, a. b,c. Let a denote a nonidentity element whose square is the identity. The row and column labeled by e are known. Show that the row labeled by a is determined by the requirement that each group element must appear exactly once in each row and column; similarly, the column labeled by a is determined. There are now four table...

4.- Show the solution: a.- Let G be a group, H a subgroup of G and...

4.- Show the solution: a.- Let G be a group, H a subgroup of G and a∈G. Prove that the coset aH has the same number of elements as H. b.- Prove that if G is a finite group and a∈G, then |a| divides |G|. Moreover, if |G| is prime then G is cyclic. c.- Prove that every group is isomorphic to a group of permutations. SUBJECT: Abstract Algebra (18,19,20)

Show that the normalizer of a 5-Sylow subgroup of A_5 has order 10, and is a...

Show that the normalizer of a 5-Sylow subgroup of A_5 has order 10, and is a maximal subgroup of A_5.

(Modern Algebra) Show that if G is a finite group that has at most one subgroup...

(Modern Algebra) Show that if G is a finite group that has at most one subgroup for each divisor of its order then G is cyclical.

In each of the following, show that ? is a subgroup of ?. (1) 1. ?...

In each of the following, show that ? is a subgroup of ?. (1) 1. ? = 〈F(R), +〉, ? = {? ∈ F(R): ?(?)=0 for every ?∈[0,1]}. 2. ? = 〈F(R), +〉, ? = {? ∈ F(R): ?(?)=−?(?)}.

G is a group and H is a normal subgroup of G. List the elements of...

G is a group and H is a normal subgroup of G. List the elements of G/H and then write the table of G/H. 1. G=Z10, H= {0,5}. (Explain why G/H is congruent to Z5) 2. G=S4 and H= {e, (12)(34), (13)(24), (14)(23)

Find a p-Sylow subgroup for each of the given groups, and prime p: a. In Z24...

Find a p-Sylow subgroup for each of the given groups, and prime p: a. In Z24 a 2-sylow subgroup b. In S4 a 2-sylow subgroup c. In A4 a 3-sylow subgroup

show that if H is a p sylow subgroup of a finite group G then for...

show that if H is a p sylow subgroup of a finite group G then for an arbitrary x in G x^-1 H x is also a p sylow subgroup of G

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