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Recent questions tagged group-theory

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What are the generators for the group G having multiplication modulo 6 as an operation?
What are the generators for the group G={1,2,3,4,5,6} having multiplication modulo 6 as an operation?
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TIFR2010-Maths-B-9
Let $G=\left \{ z \in \mathbb{C} \mid z^n = 1 \text{ for some positive integer } n \right \}$. Then under multiplication of complex numbers,$G$ is a ... order, but every element of $G$ has finite order$G$ is a cyclic groupNone of the above
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TIFR2010-Maths-A-15
Let G be the set of all 2 x 2 symmetric, invertible matrices with real entries. Then with matrix multiplication, G is.An infinite groupA finite groupNot a groupAn abelian group
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TIFR2010-Maths-A-2
Which of the following is false?Any abelian group of order 27 is cyclicAny abelian group of order 14 is cyclicAny abelian group of order 21 is cyclicAny abelian group of order 30 is cyclic
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TIFR2010-Maths-A-1
A cyclic group of order 60 has12 Generators15 Generators16 Generators20 Generators
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maths_mocktest1_30
Let $G$ be a finite group. If $A$ and $B$ are subgroups of $G$ with orders 4 and 5 respectively, then $|A \cap B|$ = ______.
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Comment on (R,*) group / commutative / monoid
Which of the following true about (R,*)?1) Group but not commutative2) A commutative group3) Not a semigroup4) Not a monoid
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GATE CSE 1996 | Question: 2.4
Which one of the following is false?The set of all bijective functions on a finite set forms a group under function compositionThe set $\{1, 2, \dots p-1\}$ forms a group under ... for any pair of elements $a, b \in S, a * b^{-1} \in S$
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GATE CSE 1996 | Question: 1.4
Which of the following statements is FALSE?The set of rational numbers is an abelian group under additionThe set of integers in an abelian group ... set of real numbers excluding zero is an abelian group under multiplication
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GATE CSE 1995 | Question: 21
Let $G_1$ and $G_2$ be subgroups of a group $G$.Show that $G_1 \cap G_2$ is also a subgroup of $G$.Is $G_1 \cup G_2$ always a subgroup of $G$?.
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GATE CSE 1995 | Question: 2.17
Let $A$ be the set of all non-singular matrices over real number and let $*$ be the matrix multiplication operation. Then$A$ is closed under $*$ ... not a group.$\langle A, *\rangle$ is a a group but not an abelian group.
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GATE CSE 1994 | Question: 2.2
On the set $N$ of non-negative integers, the binary operation ______ is associative and non-commutative.
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GATE CSE 1994 | Question: 1.10
Some group $(G, o)$ is known to be abelian. Then, which one of the following is true for $G$?$g=g^{-1} \text{ for every } g \in G$g=g^2 \text{ for every }g \in G$(goh)^2 = g^2oh^2 \text{ for every } g, h \in G$G$ is of finite order
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GATE CSE 1993 | Question: 28
Let $\left(\{ p,q \},*\right)$ be a semigroup where $p*p=q$. Show that:$p*q=q*p$ and$q*q=q$
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GATE CSE 1997 | Question: 3.1
Let $\left(Z, *\right)$ be an algebraic structure where $Z$ is the set of integers and the operation $*$ is defined by $n*m = \max(n,m)$. Which of the ... $\left(Z, *\right)$ is a groupNone of the above
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GATE CSE 2014 Set 3 | Question: 50
There are two elements $x,\:y$ in a group $(G,*)$ such that every element in the group can be written as a product of some number of $x$ ... where $e$ is the identity element. The maximum number of elements in such a group is ____.
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GATE CSE 2014 Set 3 | Question: 3
Let $G$ be a group with $15$ elements. Let $L$ be a subgroup of $G$. It is known that $L \neq\ G$ and that the size of $L$ is at least $4$. The size of $L$ is __________.
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GATE CSE 1998 | Question: 12
Let $(A, *)$ be a semigroup, Furthermore, for every $a$ and $b$ in $A$, if $a \neq b$, then $a*b \neq b*a$.Show that for every $a$ in $A$, $a*a=a$Show that for every $a$, $b$ in $A$, $a*b*a=a$Show that for every $a,b,c$ in $A$, $a*b*c=a*c$
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GATE CSE 1999 | Question: 4
Let $G$ be a finite group and $H$ be a subgroup of $G$. For $a \in G$, define $aH=\left\{ah \mid h \in H\right\}$.Show that $|aH| = |bH|.$Show ... $bH$ are disjoint.Use the above to argue that the order of $H$ must divide the order of $G.$
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GATE CSE 2007 | Question: 21
How many different non-isomorphic Abelian groups of order $4$ are there?$2$3$4$5$
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GATE CSE 2005 | Question: 46
Consider the set $H$ of all $3 * 3$ ... $H$ is:a groupa monoid but not a groupa semi group but not a monoidneither a group nor a semi group
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GATE CSE 2005 | Question: 13
The set \(\{1, 2, 4, 7, 8, 11, 13, 14\}\) is a group under multiplication modulo $15$. The inverses of $4$ and $7$ are respectively:$3$ and $13$2$ and $11$4$ and $13$8$ and $14$
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GATE CSE 2010 | Question: 4
Consider the set $S = \{1, ω, ω^2\}$, where $ω$ and $ω^2$ are cube roots of unity. If $*$ denotes the multiplication operation, the structure $(S, *)$ formsA GroupA RingAn integral domainA field
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GATE CSE 2004 | Question: 72
The following is the incomplete operation table of a $4-$ ... e\; b$c\; b\; a\; e$c\; b\; e\; a$c\; e\; a\; b$
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GATE CSE 2003 | Question: 7
Consider the set $\Sigma^*$ of all strings over the alphabet $\Sigma = \{0, 1\}$. $\Sigma^*$ with the concatenation operator for stringsdoes not form ... a right identity elementforms a group if the empty string is removed from $\Sigma^*$
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GATE CSE 2006 | Question: 3
The set $\{1,2,3,5,7,8,9\}$ under multiplication modulo $10$ is not a group. Given below are four possible reasons. Which one of them is false?It is not ... $3$ does not have an inverse$8$ does not have an inverse
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GATE CSE 2002 | Question: 1.6
Which of the following is true?The set of all rational negative numbers forms a group under multiplication.The set of all non-singular matrices forms a group ... of all matrices forms a group under multiplication.Both B and C are true.
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GATE CSE 2009 | Question: 22
For the composition table of a cyclic group shown below: ... ?$a,b$ are generators$b,c$ are generators$c,d$ are generators$d,a$ are generators
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GATE CSE 2009 | Question: 1
Which one of the following is NOT necessarily a property of a Group?CommutativityAssociativity Existence of inverse for every element Existence of identity
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GATE CSE 2000 | Question: 4
Let $S= \{0, 1, 2, 3, 4, 5, 6, 7\}$ and $⊗$ denote multiplication modulo $8,$ that is, $x ⊗ y= (xy) \mod 8$Prove that $( \{ 0, 1\}, ⊗)$ is not a group.Write three distinct groups $(G, ⊗)$ where $G ⊂ S$ and $G$ has $2$ elements.
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