Commutativity and group classification

Questions asking whether a group is abelian/commutative, or to classify/identify the type of group (e.g., distinguishing between groups of the same order).

2 questions · Challenging +1.3

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OCR FP3 2009 January Q1
5 marks Standard +0.8
1 In this question \(G\) is a group of order \(n\), where \(3 \leqslant n < 8\).
  1. In each case, write down the smallest possible value of \(n\) :
    1. if \(G\) is cyclic,
    2. if \(G\) has a proper subgroup of order 3,
    3. if \(G\) has at least two elements of order 2 .
    4. Another group has the same order as \(G\), but is not isomorphic to \(G\). Write down the possible value(s) of \(n\).
Pre-U Pre-U 9795/1 2018 June Q10
10 marks Challenging +1.8
  1. Let \(G\) be a group of order 10. Write down the possible orders of the elements of \(G\) and justify your answer. [2]
  2. Let \(G_1\) be the cyclic group of order 10 and let \(g\) be a generator of \(G_1\) (that is, an element of order 10). List the ten elements of \(G_1\) in terms of \(g\) and state the order of each element. [4]
  3. The group \(G_2\) is defined as the set of ordered pairs \((x, y)\), where \(x \in \{0, 1\}\) and \(y \in \{0, 1, 2, 3, 4\}\), together with the binary operation \(\oplus\) defined by $$(x_1, y_1) \oplus (x_2, y_2) = (x_3, y_3),$$ where \(x_3 = x_1 + x_2\) modulo 2 and \(y_3 = y_1 + y_2\) modulo 5.
    1. List the elements of \(G_2\) and state the order of each element. [3]
    2. State, with justification, whether \(G_1\) and \(G_2\) are isomorphic. [1]