Modern
Basics, Groups and Subgroups¶
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Let \(m_{1}, m_{2}, \cdots, m_{k}\) be positive integers and \(d>0\) the greatest common divisor of \(m_{1}, m_{2}, \cdots, m_{k}\). Show that there exist integers \(x_{1}, x_{2}, \cdots, x_{k}\) such that
1b 2013¶
Give an example of an infinite group in which every element has finite order.
1a 2012¶
- (a) How many elements of order 2 are there in the group of order 16 generated by \(a\) and \(b\) such that the order of \(a\) is 8 , the order of \(b\) is 2 and \(b a b^{-1}=a^{-1}\).
1a 2011¶
- (a) Show that the set
is a non-abelian group of order 6 w.r.t. composition of mappings. \(\&\)
4a 2011¶
- (a) Let a and b be elements of a group, with \(\mathrm{a}^{2}=\mathrm{e}\), \(b^{6}=e\) and \(a b=b^{4} a\)
Find the order of \(a b\), and express its inverse in each of the forms \(a^{m} b^{n}\) and \(b^{m} a^{n}\). 20
1a 2010¶
(a) Let \(G=\mathbb{R}-\{-1\}\) be the set of all real numbers omitting -1 . Define the binary relation \(*\) on \(G\) by \(a * b=a+b+a b\). Show \((G, *)\) is a group and it is abelian 12
1a 2020 IFoS¶
Let \(p\) be a prime number. Then show that
Also, find the remainder when \(6^{44} \cdot(22) !+3\) is divided by 23 .
3a 2019 IFoS¶
If in the group \(G, a^{5}=e, a b a^{-1}=b^{2}\) for some \(a, b \in G\), find the order of \(b\).
1a 2017 IFoS¶
- (a) Prove that every group of order four is Abelian.
2a 2017 IFoS¶
- (a) Let \(G\) be the set of all real numbers except -1 and define \(a * b=a+b+a b\) \(\forall a, b \in G\). Examine if \(G\) is an Abelian group under *.
1a 2016 IFoS¶
1.(a) Prove that the set of all bijective functions from a non-empty set \(X\) onto itself is a group with respect to usual composition of functions.
1a 2014 IFoS¶
(a) If \(G\) is a group in which \((a \cdot b)^{4}=a^{4} \cdot b^{4},(a \cdot b)^{5}=a^{5} \cdot b^{5}\) and \((a \cdot b)^{6}=a^{6} \cdot b^{6}\), for all \(a, b \in G\), then prove that \(G\) is Abelian.
1b 2013 IFoS¶
(b) Prove that if every element of a group \((G, 0)\) be its own inverse, then it is an abelian group.
1a 2011 IFoS¶
(a) Let \(G\) be a group, and \(x\) and \(y\) be any two elements of \(G\). If \(y^{5}=e\) and \(y x y^{-1}=x^{2}\), then show that \(O(x)=31\), where \(e\) is the identity element of \(G\) and \(x \neq e\).
1a 2010 IFoS¶
(a) Let
Show that ' \(G\) is a group under matrix multiplication.
3a 2010 IFoS¶
(a) Show that zero and unity are only idempotents of \(Z_{n}\) if \(n=p^{r}\), where \(p\) is a prime.
Cyclic Groups¶
2 a 2020
Let \(G\) be a finite cyclic group of order \(n\). Then prove that \(G\) has \(\phi(n)\) generators (where \(\phi\) is Euler's \(\phi\)-function).
2b 2016¶
Let \(p\) be a prime number and \(\mathbf{Z}_{\mathrm{p}}\) denote the additive group of integers modulo p. Show that every non-zero element of \(\mathbf{Z}_{p}\) generates \(\mathbf{Z}_{\mathbf{p}}\).
1a 2015¶
How many generators are there of the cyclic group \(G\) of order 8 Explain.
Taking a group \(\{e, a, b, c\}\) of order 4 , where \(e\) is the identity, construct composition tables showing that one is cyclic while the other is not.
1e 2011¶
(e) (i) Prove that a group of Prime order is abelian.
(ii) How many generators are there of the cyclic group \((\mathrm{G}, \cdot)\) of order 8 ?
2a 2011¶
- (a) Give an example of a group \(\mathrm{G}\) in which every proper subgroup is cyclic but the group itself is not cyclic.
1b 2010¶
(b) Show that a cyclic group of order 6 is isomorphic to the product of a cyclic group of order 2 and a cyclic group of order 3. Can you generalize this? ? Justify.
1b 2009¶
(b) Determine the number of homomorphisms from the additive group \(\mathbb{Z}_{15}\) to the additive group \(\mathbb{Z}_{10}\). ( \(\mathbb{Z}_{n}\) is the cyclic group of order \(n\) ). \(\quad 12\)
3b 2020 IFoS¶
Let \(G\) be a finite group and let \(p\) be a prime. If \(p^{m}\) divides order of \(G\), then show that \(G\) has a subgroup of order \(p^{m}\), where \(m\) is a positive integer.
2b 2016 IFoS¶
2.(b) Let \(\bar{G}\) be a group of order \(p q\), where \(p\) and \(q\) are prime numbers such that \(p>q\) and \(q \nmid(p-1)\). Then prove that \(G\) is cyclic.
1a 2015 IFoS¶
Q1. (a) If in a group \(G\) there is an element \(a\) of order 360 , what is the order of \(a^{220}\) ? Show that if \(G\) is a cyclic group of order \(\mathrm{n}\) and mivides \(\mathrm{n}\), then \(G\) has a subgroup of order \(m\).
3b 2011 IFoS¶
(b) Let \(G\) be a group of order \(2 p, p\) prime. Show that either \(G\) is cyclic or \(G\) is generated by \(\{a, b\}\) with relations \(a^{p}=e=b^{2}\) and \(b a b=a^{-1}\).
Normal Subgroups¶
Homomorphism¶
2a 2019¶
If \(G\) and \(H\) are finite groups whose orders are relatively prime, then prove that there is only one homomorphism from \(G\) to \(H\), the trivial one.
2a 2018¶
Show that the quotient group of \((\mathbb{R},+)\) modulo \(\mathbb{Z}\) is isomorphic to the multiplicative group of complex numbers on the unit circle in the complex plane. Here \(\mathbb{R}\) is the set of real numbers and \(\mathbb{Z}\) is the set of integers.
2a 2019¶
If \(G\) and \(H\) are finite groups whose orders are relatively prime, then prove that there is only one homomorphism from \(G\) to \(H\), the trivial one.
2a 2018¶
Show that the quotient group of \((\mathbb{R},+)\) modulo \(\mathbb{Z}\) is isomorphic to the multiplicative group of complex numbers on the unit circle in the complex plane. Here \(\mathbb{R}\) is the set of real numbers and \(\mathbb{Z}\) is the set of integers.
3a 2017¶
Show that the groups \(\mathbb{Z}_{5} \times \mathbb{Z}_{7}\) and \(\mathbb{Z}_{35}\) are isomorphic.
2a 2010¶
Let \(\left(\mathbb{R}^{*}\right.\), \()\) be the multiplicative group of nonzero reals and \((G L(n, \mathbb{R}), X)\) be the multiplicative group of \(n \times n\) non-singular' real matrices. Show that the quotient group \(G L(n, \mathbb{R}) / S L(n, \mathbb{R})\) and \(\left(\mathbb{R}^{*}, \cdot\right)\) are isomorphic where
\(S L(n, \mathbb{I})=\{A \in G L(n, \mathbb{I R}) / \operatorname{det} A=1\}\).
What is the centre of \(G L(n\), IR) ?
1a 2009¶
If \(\mathbb{R}\) is the set of real numbers and \(\mathbb{R}_{+}\)is the set of positive real numbers, show that \(\mathbb{R}\) under addition \((\mathbb{R},+)\) and \(\mathbb{R}\), under multiplication \(\left(\mathbb{R}_{+}, \cdot\right)\) are isomorphic. Similarly if \(\mathbb{Q}\) is the set of rational numbers and \(Q_{+}\)the set of positive rational numbers, are \((\mathbb{Q},+)\) and \(\left(\mathbb{Q}_{+}, \cdot\right)\) isomorphric? Justify your answer. \(4+8=12\)
2a 2018 IFoS¶
Find all the homomorphisms from the group \((\mathbb{Z},+)\) to \(\left(\mathbb{Z}_{4},+\right)\).
2a 2011 IFoS¶
Let \(G\) be the group of non-zero complex numbers under multiplication, and let \(N\) be the set of complex numbers of absolute value 1 . Show that \(G / N\) is isomorphic to the group of all positive real numbers under multiplication.
2b 2010 IFoS¶
Prove or disprove that \((\mathbb{R},+)\) and \(\left(\mathbb{R}^{+}, \cdot\right)\) are isomorphic groups where \(\mathbb{R}^{+}\) denotes the set of all positive real numbers.