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Show that an element of a group has order n if and only if it generates a cyclic group of order n?
An element ( g ) of a group ( G ) has order ( n ) if the smallest positive integer ( k ) such that ( g^k = e ) (the identity element) is ( n ). This means the powers of ( g ) generate the set ( { e, g, g^2, \ldots, g^{n-1} } ), which contains ( n ) distinct elements. Therefore, the cyclic group generated by ( g ), denoted ( \langle g \rangle ), has exactly ( n ) elements, thus it is a cyclic group of order ( n ). Conversely, if ( \langle g \rangle ) is a cyclic group of order ( n ), then ( g ) must also have order ( n ) since ( g^n = e ) is the first occurrence of the identity.
How can you find number of homomorphism between two rings?
No of homo from the group zm to the group zn is gcd(m,n) No of homo from the ring zm to the ring zn is 2^(w(n)-w(n/gcd(m,n)))
What is the mathematical formula for the number of possible relationships in a group?
N * (n-1 ) / 2
What is the automorphism group of a complete graph?
The automorphism group of a complete graph ( K_n ) (where ( n ) is the number of vertices) is the symmetric group ( S_n ). This is because any permutation of the vertices of ( K_n ) results in an isomorphic graph, as all vertices are equivalent in a complete graph. Therefore, the automorphism group consists of all possible ways to rearrange the vertices, corresponding to the ( n! ) permutations of the ( n ) vertices.
What number can you add to 7 that can group into 4 so the answer is 6?
n + 7 = 4*6 n + 7 = 24 n = 24 - 7 = 17.
Fidelitys rock n roll group where did this group form from and who are the singers in the group?
where did the fidelitys rock n roll group form? Albany, N.Y.
What are the group configuration notation for each d block group?
group 3- (n-1)d1 ns2. Group 12 (n-1)d10 ns2, groups 4-11 do not necessarily have identical outer electron configurations. Where n represents your period that you are in.
Are n-dubz a British group?
yes N-dubz is a British group they are from north London
What is an azadipeptide?
An azadipeptide is any of a group of aromatic dipeptides which have an N-N group in place of the terminal amine.
If a 2-person group equals 1 dyad and a 3-person group equals 3 dyads then what is the formula for determining the no. of dyads in group of N members?
n*(n-1)/2
What are the symbols of permutation groups?
some examples of symbols for permuation groups are: Sn Cn An These are the symmetric group, the cyclic group and the alternating group of order n. (Alternating group is order n!/2, n>2) One other is the Dihedral group Dn of order 2n.
Show that an element of a group has order n if and only if it generates a cyclic group of order n?
An element ( g ) of a group ( G ) has order ( n ) if the smallest positive integer ( k ) such that ( g^k = e ) (the identity element) is ( n ). This means the powers of ( g ) generate the set ( { e, g, g^2, \ldots, g^{n-1} } ), which contains ( n ) distinct elements. Therefore, the cyclic group generated by ( g ), denoted ( \langle g \rangle ), has exactly ( n ) elements, thus it is a cyclic group of order ( n ). Conversely, if ( \langle g \rangle ) is a cyclic group of order ( n ), then ( g ) must also have order ( n ) since ( g^n = e ) is the first occurrence of the identity.
What is the group number for Terbium?
Terbium has a group number of 3.
What group is tulisa in?
N-dubz
What group ruled after Ali?
N
Who is the CEO of TATA Group?
The CEO of TATA Group is Mr. N. Chandrashekaran.
What name is used to describe H-N-H group?
amine group
