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Definition
| a finite abelian group that is not cyclic |
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| does not exist since the subgroups of a cyclic group must be cyclic |
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Definition
| a finite cyclic group with a noncyclic subgroup |
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| matrix multiplication on 2X2 real matrices |
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Definition
| a binary operation on a finite set that is not commutative |
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Definition
| a group ℤn where 3 and 7 are inverses of each other |
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Definition
| a group ℤn where 4 and 5 are both generators of the group but not inverses of each other |
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Definition
| a non-abelian group of order 24 |
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Term
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Definition
| an infinite non-cyclic, abelian group |
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Term
| GL(2,R) = group of 2X2 invertible matrices with matrix multiplication |
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Definition
| an infinite non-abelian group |
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Term
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Definition
| a finite non-abelian group where all proper subgroups are cyclic |
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Definition
| a finite abelian group where all of its proper subgroups are cyclic but the group itself is not cyclic |
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Term
| ℤ20 where <4> = {0,4,8,12,16} so |<4>| = 5 |
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Definition
| a finite cyclic group G that has a subgroup H = <4> and |H| = 5 |
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Term
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Definition
| infinite non-cyclic group that is finitely generated |
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Definition
| a finite non-abelian group where all subgroups are normal |
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Definition
| a cyclic subgroup of a group that is not normal |
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ϕ: GL(2, R) -> R*
where ϕ(A) = det(A) |
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Definition
| a homomorphism from a non-abelian group to an abelian group |
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Term
| the subset { p0, u1, u2, u3 } in S3 is not a subgroup |
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Definition
| a subset of a group that contains the identity and all elements of order 2 where the subset is not a subgroup |
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Term
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Definition
| a finite cyclic group that is decomposable |
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Term
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Definition
| a finite cyclic group that is not decomposable |
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G = ℤ and H = 5ℤ
with addition as operations |
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Definition
| a subgroup H of a group G where both H and G are infinite but [G:H] = 5. Specify both G and H. |
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Term
G = S3 and H = < p1 >
since [G:H] = 2 so H <- G and G/H ~ ℤ2 |
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Definition
| a factor group G/H that is isomorphic to ℤ2 where G and H are both finite and G is non-abelian |
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Term
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Definition
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Definition
| a finite ring with zero divisors |
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M2(R)
i.e. 2X2 real matrices with matrix addition and multiplication |
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Definition
| an infinite ring with zero divisors |
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Term
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Definition
| an even (non-trivial) element from S4 written as a product of disjoint cycles |
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Term
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Definition
| an odd (non-trivial) element from S4 written as a product of disjoint cycles |
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not possible
i.e. ℤ15 ~ ℤ3 X ℤ5 but this is cyclic |
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Definition
| an abelian group of order 15 that is not cyclic |
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Definition
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Definition
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Definition
| Ring with non-commutative multiplication |
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Definition
| Finite ring with zero divisors |
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